You don't need many supplements at all to be successful in improving your fitness. In fact, you don't need any if you're eating well enough. Sometimes, however, life doesn't permit you to be as diligent as you'd like with your eating, so you might see some benefits from supplementing your diet. This doesn't mean you can neglect proper eating just because you take a protein powder and a multivitamin, but there's nothing wrong with adding in a few things that might help you out.
Here are some supplements that I think are worthwhile. After, in part 2, I'll discuss the supplements that are, at best, terrible ways to spend money that you could be using for real people things, and at worst, downright dangerous.
Worthwhile Supplements
Caffeine
Caffeine has been shown to reduce fatigue, increase blood pressure and pulse rate in the short term, improve endurance (particularly by allowing you to sustain more forceful lower-intensity muscle contractions for an extended period), improve mental focus, trigger release of gastrin (which stimulates gastric motility), and increase the metabolism of bodyfat as energy (1, 2). What this means for working out is that caffeine can help get you through long, voluminous workouts and maintain a higher intensity throughout, and be less prone to distraction. It can also perk you up when you feel tired, help you burn fat more efficiently, and make you poop. The last part is relevant because if you take caffeine with a full colon right before a workout, you may find yourself exploding in the bottom of a squat, so tread carefully.
Something that should also be mentioned with caffeine is that the more you take it, the more you'll build a tolerance to it, so you'll need more to get an effect. Additionally, if you use it enough you can become dependent on it, and without it you may experience withdrawal symptoms. I personally only ever have caffeine in the form of coffee, and I don't drink it more than once or twice a week at most to keep my tolerance low. On a somewhat related note, coffee itself has been shown to have a number of health benefits, and an often-cited study found that coffee drinkers may live longer (3, 4)
Whey Protein Powder
This is probably the first supplement a person will buy when they start working out, and for good reason - it's excellent to take right around a workout. Whey is one of the two primary proteins that make up milk, and is a very fast-digesting and complete protein that contains branched chain amino acids (BCAAs), which can help prevent muscle catabolism if taken pre-workout, and help stimulate muscle growth and repair after a workout. Leucine, which whey contains a significant amount of, is a particularly good trigger of anabolic pathways (5). Ingestion of whey before, during, and immediately after a workout is an easy step that can be immediately taken to get the most out of your workouts. Because whey is digested so quickly, however, it is a poor protein to use as a meal replacement and will leave you hungry soon after taking it. For that, the next supplement is far better.
Casein Protein Powder
Casein is the other primary protein in milk, and it's basically whey's polar opposite in that it is extremely slow-digesting. Because of this, casein is perfect to use when you won't be able to get in any protein for a while. When amino acids aren't being released into the bloodstream from digesting proteins, your body can catabolize its own muscle to supply sufficient amino acids for bodily processes. So if you aren't taking in protein for a 6+ hour period, you're doing yourself a disservice. Casein takes up to 8 or so hours to fully digest, so it's a perfect protein to take before bed or when you know you won't be getting any meals in for a while, but for this same reason it's definitely not the best protein to consume directly after a workout (6). When I have 7 hours of classes and don't feel like packing a bunch of food in my backpack, I'll often just bring a scoop of casein to add water to and drink later, since it's quick and easy to consume, high in protein, and won't perish if you keep it dry. It's more important to keep a relatively steady intake of protein than of carbs or fat, so this should be a priority.
For both casein and whey, since they're both the main proteins in milk, you could just as well consume some milk and get many of the same benefits, but if you want to functionally isolate each protein in a more concentrated amount or avoid the carbs and fats of milk, I'd recommend buying the powders. You can also get a blended powder that will contain whey, casein, and possibly some other proteins as well for a more "hybrid" effect.
Also, as far as both powders go, I'd recommend any isolate that you think tastes the best. All isolates are basically the same. Concentrates aren't awful either, they're just less pure so you need to use more powder to get the same amount of protein. Hydrolyzed whey is overpriced for the small marginal benefits it may possibly offer, so I wouldn't suggest buying it.
Gatorade Powder
Ok, Gatorade isn't really a "supplement" per se, but you can use it like one. There is evidence that consuming a fast-digesting carbohydrate before, during, and after training can help enhance training quality during hard sessions and maximize the muscle-building results of the workout. Sipping a fast-carb drink before and during a particularly long or voluminous workout can help assure you're provided with a steady supply of glucose which can provide you with sufficient energy to make it through. It can also help decrease the chances of muscle catabolism during the workout slightly. Consuming some fast carbs after training can help muscle building by repleting glycogen stores and activating anabolic pathways (7). Gatorade fits the profile of "fast carbs" perfectly.
The reason I suggest buying powder is twofold. One, it is exponentially cheaper than buying individual bottles. Two, you can adjust the concentration to whatever you want. A more diluted solution will most likely be more conducive to absorption.
You can combine Gatorade and whey powder into a pre-, during-, or post-workout shake so that you're getting the necessary nutrients in through a convenient single drink. You can either use unflavored whey, so that the drink will only take on the taste of the Gatorade (with a slightly creamier taste and texture... it's not as bad as it sounds), or you can combine flavored Gatorade with flavored whey. I'm sure there are a lot of combinations that would taste good. I've been taking fruit punch Gatorade with chocolate whey at the suggestion of Renaissance Periodization's Nick Shaw, and it's surprisingly good. It actually tastes just like a red Tootsie Pop.
Hell yeah.
ZMA
ZMA stands for zinc, magnesium, and aspartate (it also contains vitamin B6). You're supposed to take it about an hour before bed, and it's supposed to improve sleep quality and hormone production (particularly of growth hormone) while you sleep to enhance recovery ability. There isn't any particularly substantial clinical evidence to back this up, but I've been taking it for a while and I do think it has had a positive impact on my sleep quality and recovery abilities. It's important to note that it isn't meant to induce sleep, just to improve sleep quality and hormone secretions throughout the night, so take it for what it's worth.
Probiotics
If you have digestive issues, you should be taking a probiotic. For those who don't know, probiotics are orally-delivered doses of helpful bacteria such as lactobacilli that aid in digestion of food, particularly in the colon. Your colon is naturally populated with symbiotic bacteria, and probiotics can help make sure that the right kind of bacteria out-compete the wrong kinds. The nature of this symbiotic relationship is essentially that you provide the bacteria with nutrition through what you ingest so that they can grow and reproduce, and they provide you with the benefit of improved digestive health. I don't think probiotics are a cure-all like some people have recently been saying, but they're definitely worth trying.
Probiotics can come in the form of a chewable tablet, a capsule, or simply in foods such as yogurt or kefir. I believe that taking a probiotic was a huge part in helping me adapt to eating Rutgers dining hall food, which destroyed my digestive tract for the first several months of college. Probiotics would be particularly beneficial if you are eating a lot, such as if trying to gain weight, or if you are just coming off of an antibiotic regimen, as some antibiotics might kill your helpful gut bacteria in addition to the pathogens that they're intended to kill.
Creatine
Creatine works. It's not a steroid, it's not going to make you super jacked or strong overnight, but it can have positive effects on muscle synthesis and performance. Your body utilizes several different energy systems in a particular order based on the duration of activity. First it'll use existing ATP from the muscles and creatine phosphate for activity of any intensity for 10-15 seconds. For intense activities such as sprinting which cannot be sustained for a long period of time, this is the system that will usually be used. After the ATP-PC stores are exhausted, energy will be taken from muscle glycogen stores, then liver glycogen stores, and finally from oxidation of fatty acids from adipose tissue. In the energy systems following ATP-PC, processes begin as anaerobic and eventually move on to oxidative aerobic systems.
The point of creatine is that it can donate a phosphate group to lower-energy ADP (adenosine diphosphate) to synthesize the high-energy ATP (adenosine triphosphate), which is directly used for energy. The ATP-PC system can recover after a fairly short period of rest, which is why creatine supplementation can give you the ability to perform more high-intensity sets in a given workout. Supplementation might also increase the duration of ATP-PC activity, which means that you might be able to push out a couple of extra reps while on creatine (8).
Creatine is synthesized naturally in the body, and also is provided in meats. A "loading phase" where you take larger than normal doses for a short time to saturate your muscles with creatine is often recommended on the bottle, and it may or not be effective. I didn't do a loading phase when I started using it, and I don't think its effectiveness was negatively impacted. Whether you do the loading phase or not is up to you.
As for which creatine to buy, I'd say to just buy whichever monohydrate is the cheapest. They're all pretty much the same, and it's not worth spending big money on one that claims to be higher quality than standard creatine monohydrate.
Fish Oil
I've been taking fish oil for about 4 years now, and while there is somewhat conflicting evidence as to how effective it really is in preventing heart disease and some of its other alleged benefits, I'd suggest it purely based on the fact that I think it's helped improve the appearance of my skin. Purely anecdotally, I noticed that my skin became a lot clearer once I started taking it consistently. I don't attribute this entirely to fish oil, but I do believe it played a role.
As for its alleged abilities to prevent heart disease and improve recovery abilities, I'm not entirely convinced, but it certainly can't hurt your heart health or recovery. Having extra healthy fats might be able to "lubricate" the joints and muscles to help you move more safely, as well as reduce chronic inflammation. It's also been thought that regular consumption of fish may be correlated to improved cognition later in life relative to non-consumers (9). Consuming fish oil rather than whole fish has the benefit of avoiding mercury consumption, which is never a good thing (though you'd need to consume a LOT of apex predator fish to actually get mercury poisoning). Since fish oil can be dirt cheap if you're buying the right brand, it's probably a good idea to give it a shot just for the hell of it.
I'd recommend buying the extremely reasonable Sundown Naturals brand, and buying the largest container you can find for value's sake.
Bad supplements coming at you in part 2.
References
1. http://www.vanderbilt.edu/AnS/psychology/health_psychology/caffeine_sports.htm
2. http://www.rice.edu/~jenky/sports/caffeine.html
3. http://www.nejm.org/doi/full/10.1056/NEJMoa1112010
4. http://authoritynutrition.com/top-13-evidence-based-health-benefits-of-coffee/
5. http://www.nutritionexpress.com/article+index/authors/jeff+s+volek+phd+rd/showarticle.aspx?articleid=807
6. http://www.bodybuilding.com/fun/the-case-for-casein.html
7. Israetel, Michael, Jennifer Case, and James Hoffman. The Renaissance Diet. 2014. E-book.
8. Clark, Micheal, Brian Sutton Lucett, and Scott C. Lucett. NASM Essentials of Personal Fitness Training. 4th ed. Philadelphia: Wolters Kluwer Health/Lippincott Williams & Wilkins, 2014. Print.
9. http://www.webmd.com/diet/what-to-know-about-omega-3s-and-fish?page=3