Some of these I've talked about in detail with individual posts, others I intend to get to in the not-too-distant future. Right now, I'm just going to list a few things that I think are worth looking into.
Accessories - Stuff you add to your bike, or carry with you
Multitools: You should have one and know how to do at least basic fixes and adjustments. You don't have to be a master mechanic, but knowing how to adjust your seat height is only reasonable.
Minipump: Are you going to ride farther from your house than the distance you'd like to walk? Bring an inexpensive portable pump, clipped to your frame or in your bag.
Spare Inner Tube: Goes with the pump above, it's much easier to replace the whole tube in wet or cold weather than to use a
Patch Kit: On long rides, it's worth carrying a spare tube AND a patch kit, in case of multiple flats (it happens, replace the tube first, but save the old one in case you get a new hole, that way you have to potentially patch-able tubes). Traditional patch kits work well, but can be messy. "Glueless" patch kits are compact and easy to use, but the patches may not last as long (they'll hold up for a month or two, certainly long enough to get you home).
Tire Levers: Complete your flat-fixing kit with some levers. These will help you get the tire off the rim without damaging it or yourself.
Fenders: If you're riding in anything but dry weather, at least some sort of rear mudguard will keep your rear end from getting soaked. Full fenders are worth their weight in gold if your bike will fit them.
Lights: From dusk till dawn, you need 'em. Simple clip-ons are fine for just being seen, but higher-wattage lights up front will help you see where you're going if you venture beyond the streetlights. Truly bright front lights are pricey, but new LED and battery technology means they're both brighter and cheaper than ever, and you can get a lot of Lumens for under $100. Battery power is still the way to go on a budget, but if you find yourself with a few hundred dollars extra, you could do worse than to throw it into a front wheel with a built in dynamo hub.
Racks: Rear racks are pretty ubiquitous on commuter bikes, and will let you sling bags or bungee cargo on board. A front rack isn't a bad idea if your bike will take one. A front rack will let you keep an eye on delicate cargo or give you a stable platform to attach a basket.
Baskets: The cheapest and simplest way to make a bike into a practical vehicle is to slap on a basket. Suddenly, you can carry stuff without a backpack or messenger bag, and the simple, open-top convenience of a basket means you don't even have to think twice about grabbing some groceries on the way home.
Bottle Cages: If you're riding more than a mile or two, especially in the summer, the ability to carry at least one water bottle in an easily accessible spot is wonderful.
Bell: One of the least-often enforced traffic laws in most areas is the requirement that bicycles have an "audible signaling device," such as a bell or horn. You can skip this one at your discretion, but if you ride on a multi-use path at all, where you've got walkers, joggers and slower riders, you'll get tired of yelling "excuse me" every time you want to pass. A bell will save your vocal chords a bit.
Bags (on the bike): There are any number of bicycle-mounted bags to choose from. The most common are small under-seat bags that hold your multitool and spare tube, but you can also get ones to carry touring gear, groceries, maps, laptops and whatever else you need. Size, price and mounting configuration will vary based on your needs and your budget.
Bags (on you): A backpack or messenger bag can be handy if you need to carry stuff with you once you're off the bike. If you lock up somewhere where stuff is likely to be stolen, keeping even your tools and pump in a bag is a good strategy. On-the-body bags can lead to sore shoulders on long rides, and sweaty backs on hot days, but sometimes bags can be clipped to racks or dropped into baskets to avoid this.
Lock: It's nice to come back and find your bike where you left it. A sturdy lock is a necessity for most of us.
Mirror: If you're comfortable looking over your shoulder without swerving out into traffic, you don't need a mirror on your bike. Otherwise, you'll probably feel safer using one. Even cyclists who don't normally feel the need for a mirror might benefit from one if they're riding a loaded touring bike, tandem or bike with a child seat.
Upgrades - Things already on the bike that are worth making better
Saddle: The stock saddles on most low-end bikes tend to be squishy foam things, which can cause pressure points and chafing on longer rides. You may find the factory default is good enough, but if not, this can have a huge effect on your comfort, so don't be afraid to swap for better quality.
Pedals: If they're plastic, chuck 'em. Metal-bodied pedals last much longer and can take more of a beating. My favorite pedals for commuting are big, grippy BMX-style pedals, that give you a nice supportive platform no matter what shoe type you're wearing and how big your feet may be.
Grips: If you're going to ride in wet weather often, mountain bike grips may start to come loose over time. Lock-on grips can prevent this. For long commutes, ergonomic grips are a nice, if geeky-looking option.
Wheels: Better, stronger wheels can make your bike more durable and pleasant to ride. Double-wall rims are a must for heavier riders.
Stuff you DON'T need:
Cyclometer/Bike Computer: Some folks like to know how far they've gone, or how much time they've put in, or whatever, but unless you're training for a race or for fitness, you don't NEED one of these. Personally, when the battery died on mine a couple years ago, I took it off and haven't missed it.
Smartphone Mount: I consider trying to talk on the phone while biking the same as trying to do it while driving. You need to pay attention, so turn off the ringer, toss the phone in a bag or pocket and enjoy the ride. Also, don't ride with headphones in, whether for your phone or a music player. If you don't hear that car coming behind you and slowing down, how are you going to know he's about to make a right turn in front of you?
Tires: Ride whatever is on the bike until they're bald, then replace them. Don't worry about whether they're the "perfect" tires for the season or for you local streets. As long as they're not lightweight race tires, they're fine for commuting. When they do wear out, get cheap, sturdy tires with a some sort of flat protection built in.
Tire Liners: Unless you live somewhere with a lot of cactus thorns, these "flat protection" strips are more trouble than they're worth,and often end up causing flats themselves, as they pinch against inner tubes. Get tires with a built-in flat protection strip if you can, or just don't worry about it.
Slime/Tire Sealant: This (usually green) goo claims it will prevent you from losing air from punctures. It does work, but only on very small holes and slow leaks. What it usually does in reality is ooze all over the place between the inner tube and tire and make a huge mess. Sometimes it also jams up the valves. If you have a problem with small debris puncturing your tires, you're better off using heavy-duty tubes and tires with some flat protection. I've seen tubes full of sealant leak so bad that otherwise perfectly good tires had to be thrown out with the flat inner tube, because the sticky mess was impossible to clean up.
Toe Clips and Straps: If you're racing or doing "sporty" rides, get clipless pedals, which mate to special shoes and click in like ski bindings. If you're commuting, use plain flat pedals. To get any sort of efficiency benefit from toe straps, they have to be tight enough that you'll have trouble getting out of them in a hurry. If they're loose enough to get out of easily, you're not getting any real benefit from them, but still risk wrenching your ankle if you take a spill (modern clipless pedals are designed so you'll pop out in a crash, toe straps just hold on until you tear a tendon). You don't gain much performance benefit from being attached to the pedals (it's worth it if you're racing, for commuting, not so much) and trying to pull up on the pedals to get extra power doesn't actually help at all (it costs you efficiency on the side that's pushing down) and can actually result in some pretty nasty ankle injuries (more torn tendons).
Aero Bars: If you're doing a triathlon or time trial, clip on aero bars can be helpful, otherwise, you're usually better off without them. People sometimes see them and think "oh, an elbow rest, that must make riding way more comfortable" without realizing "hey, to rest my elbows on there, I have to lean over WAAAAY further than I would usually go." Even if you are more comfortable with them, your still in a poor position to reach the brakes quickly, or to control the bike (which is why they're banned from any racing that's done in a pack, they're only legal for certain circumstances where riders are spread out, otherwise, they cause crashes).
Saddle Pads: Extra cushion that goes over a saddle, often with some sort of gel or foam. A lot of the time, these things start to slide all over the place and make the saddle feel more uncomfortable or awkward, especially cheap ones. Better quality saddle pads cost as much as an inexpensive saddle, and are still prone to sliding around. You're better off just replacing the seat if you're unhappy than trying to pile more stuff on top of it to get comfortable.