Here is a small list of suggested equipment for commuting (and/or doing errands, basic transportation, etc.) by bicycle. No one paid me for any of these endorsements.
I consider this essential -- cars are bigger than I am. Find one that fits your head, doesn't slide around or threaten to come off, is comfortable, and is certified by Snell, ANSI, or ASTM. Everyone's head is different, so I won't recommend a particular helmet for yours. I would recommend a visor, however -- helps in keeping the sun and rain out of the eyes. And apparently, the round bowl-shaped helmets are safer in crashes than the racing-style aerodynamic-looking helmets. Unless you are racing at top speed, I do not think you'll notice any extra air resistance. Here's a web site about helmet safety: helmets.org
I am lucky enough not to have to wear corrective lenses, but I find that when I'm riding my bike, I want something to protect my eyes from bugs, sand, rain, and whatever else might be flying through the air. When I'm cycling during daylight hours, I sometimes want sun glasses, but for the most part I just use clear lenses. Some people also like the yellow or orange lenses that help you see in foggy conditions; you don't want them at night, however. Ones that wrap around your head a bit are nice, because they help with peripheral vision, an essential part of bike safety. My latest find is that clear safety glasses made by 3M (available at Home Depot) are pretty comfortable.
I used to live and bicycle in Seattle, in the land of rain. I now live across the state in Spokane, which is much drier, but I still tour in places where it rains sometimes.
I think that yellow or pink cycling jackets are the most visible -- especially for riding in questionable lighting conditions. Red and blue, though perhaps more fashionable for off-bicycle, are just not as easily seen, and being seen is your first defense. Green can blend in with trees and bushes.
When I was riding an upright bike, I chose to buy a well-ventilated truly waterproof (i.e. coated fabric) Burley Rain Rider jacket rather than a "breathable/waterproof" jacket, and was quite happy with it. It kept me dry, and had a wide variety of ventilation options, including "pit zips" that cover almost the entire length of your arm and nearly all the way down your side, front Velcro closures, and a flap on the back. I'm not sure whether Burley still makes coated-fabric jackets, though, and I think the breathable/waterproof fabrics may have improved in the last decade... we can always hope! They certainly used to be failures at being waterproof, and I never thought they really breathed much better than coated fabrics either.
When shopping for a jacket, be sure to get a jacket big enough to wear extra clothes under. Check for multiple ventilation options -- front full-length zipper, upper chest slits you can open, and really long armpit zippers are good. The cycling-style jackets with the short fronts and long backs are handy for keeping you covered in cycling position, if you are riding a traditional, upright bike.... Now that I ride a recumbent bike, I have switched to just using a jacket made for hiking.
Some people like rain pants. I don't, for short trips (errands, commuting) anyway. In the summer, I just wear shorts and if I get a bit wet, I didn't care because it is warm. In the winter, I use a pair of cycling tights, or leg warmers, and the thing is that during a short trip, it's unlikely that you will get so wet as to get cold before you get to your destination, and my experience has been that in rain pants, I am usually too hot.
Gloves I think are a good idea. If you crash, they provide some level of skin protection for your hands. If you ride in cold weather, they keep your hands somewhat warmer. If you ride even in hot weather, they cut down on vibrations. Get ones with some kind of gel, if you're on an upright bicycle, and I would recommend the terry-backed ones rather than the cheap cotton mesh. Terry is better for wiping the sweat off your brow, rain off your mirror, etc. For a recumbent rider, thinner gloves are more comfortable, if you can find ones without so much gel.
Like I said earlier, I often end up riding in the rain. Fenders are wonderful at helping to keep the puddle water down where it belongs, and not heavy enough to want to avoid having them. Get plastic ones that wrap around the wheel, not the new ones that just sort of stick out straight. They take a bit longer to install, but are worth it. I don't have a particular brand to recommend -- I usually just end up getting whatever the bike shop has in stock when one cracks, which they do occasionally, mostly the front one. Maybe I should switch to metal fenders for durability?
Lights and Reflectors
If you are going to ride at night at all, it is essential to have a good lighting system, at least if you want the cars to see and avoid hitting you. Even if you don't plan to ride at night, it will come up occasionally, so it's not a bad idea to be prepared with an inexpensive but adequate lighting system.
For the back of your bicycle, your objective is to be seen by cars coming up behind you. The LED flashing rear lights are great for this. Technically, the ones that blink on and off are not legal in some places, but in the US, even where laws prohibit their use, they are generally encouraged by the police. When I bought the latest flasher for my current bicycle, I went to a bike shop that had a bunch of them set up so you could try them, and bought the one that seemed to be brightest from the most different directions (some side visibility is always a plus, and people will also be at different heights as they come up behind you). New ones come out every year, so I won't recommend a particular brand.
In the front, your objective depends on where you will be riding. You definitely want to be seen by oncoming traffic, especially cars that are coming towards you and planning to turn left in front of you (translation: read that as "planning to turn to the right", if you are in England, Austrailia, Japan, etc. where you drive on the other side of the road). Also, if you are riding on a bicycle trail or rural road, you will need sufficient light so that you can see bumps and other road obstacles in time to avoid them.
If all you need is to be seen, there are white LED front lights that are sufficiently bright, in my opinion. Get the brightest one you can find; some of the white LED lights are better than others. If you need the light for riding in areas without lights, though, I don't think the white LED lights powered by AA or AAA batteries are sufficient. You'll need to investigate a more serious lighting system with a plug-in rechargable (and fairly heavy) battery pack.
I also think reflectors are a good idea. I spent a few dollars and bought a package or two of stick-on reflectors, so now I basically have reflectors all over my helmet, fenders, and bicycle. My jacket, panniers, and cycling tights also came with reflective strips on them -- I have been told that the reflective wrist strip on the jacket was really visible when I used my arm for a turn signal. And the cheap white wheel reflectors that attach to your spokes really help in visibility from the side (which is good when a car comes up a side street towards you).
I think these are quite useful for riding in traffic. I had one called a Mirrycle for years that attaches to the brake hood of drop handlebars, and is excellent -- stable to vibrations, and wide-angle for maximum viewing. They used to be hard to find, but now the company finally has a web site, so if you want to find their products (including their really nice "Incredibell" bicycle bells), or need to order parts after a crash, you can visit: mirrycle.com.
Shoes and Pedals
I used to commute using toe clips on my pedals and touring shoes, but not any more -- I have switched to cleats. They are simply wonderful: much easier to get in and out of than the toe clips, and more of your pedaling energy gets into the drive train of the bicycle. I went with the Speedplay Frog Pedals, and I love them. I think they're easier to clip into and out of than other pedal systems (though I gather that you can get used to anything), and I especially like that you don't have to apply torque or wrench your foot to clip out when you stop, yet I have never ended up clipped out of them by mistake.
When I was still riding an upright bike, I always used Continental Touring Top 2000 tires, which are wonderful. Perhaps they have more road resistance than some people would put up with, but I think they are a good tradeoff between some traction for gravel and rain and a reasonably smooth riding surface. I had few flats, even on fully loaded tours, aside from when I ran over something sharp and large.
Now that I am riding a recumbent with very small wheels, I can no longer use these tires (they don't make them in my size), but I've been pretty happy with the Schwalbe Marathons I use now. Their Marathon Plus is also good, though it apparently has more rolling resistance.
Rack and Panniers
I see people commuting with backpacks, and worse yet, bags they carry over one shoulder. Maybe they are happy with them, but my opinion is that they just haven't experienced the joy of riding a bike with nothing weighing down their backs. And now that I've switched to a recumbent, a backpack or bag is clearly not an option.
You'll need a strong rack to carry your bag(s) on. Don't buy a cheap rack, or it may break at a bad moment.
Use plastic bags inside your panniers, unless you get the ones that are made like kayaking bags. Practically nothing else is waterproof to splashes coming off cars.
Someone wrote to me to suggest using a bicycle suit bag -- if you need to dress up in wrinkle-free business attire when you arrive at work, this could be a good option. Or else, the person who suggested it could be an employee of the company (TwoWheelGear) that makes them (grin!). I'ver personally never tried them.
I ride in all kinds of weather, through gravel areas, etc. So, I want to have a chain lubricant that will hold up to that kind of abuse. I've tried quite a few. For quite a while, I used White Lightning, but I got tired of all the waxy build-up it left on everything (even though it came highly recommended from my favorite local bike shop -- maybe it works really well in less extreme conditions). So, I've switched to an even more high-tech one, called "Dumonde Tech". Expensive. Great.