This blog series is going to explain to my advice on being an urban cyclist. The difficulty I've seen with other people is that while a lot of people know how to ride a bike, they may not feel comfortable riding in heavy traffic, on poorly paved roads, or in poorly lit ares. These posts are based on my experience the last six or seven years of my life cycling mostly around Berkeley, Oakland, San Francisco, and Los Angeles.
The first post in the series will cover what kind of bicycle I recommend, and what kind of gear you need to ride.
First you should have a well maintained bicycle. If you're buying a new bicycle, I strongly recommend getting a road bike with drop bars rather than a cheapo mountain bike. Road bikes are simply a lot faster, and if you don't feel fast you're not going to want to bike. There's nothing more frustrating than seeing people whiz by you on their bikes while you're struggling on yours. Simply put: if you don't feel good on your bike, you're not going to use it.
You can get a decent used steel frame bicycle in the Bay Area for $500-$600 or cheaper, depending on exactly what size frame you need, what type of components you want, etc. If you live elsewhere, you can probably get one cheaper. A decent new road bike will be something like $1000 or more if you want to get really fancy. If you're buying a new bike, I'm a big fan of Surly Bikes, but there's nothing wrong with getting a used bike. If you get a used bike, make sure you ride it and test that it can shift smoothly and brake quickly.
You should get and wear a helmet. You'll easily exceed speeds of 20 mph on your bike, and even in dense urban areas cars frequently exceedp speeds of 30 mph or more. For a comparison, falling off the top of a two story building entails an impact of about 20 mph. At 20 mph, much less at higher speeds, you can very easily die in a head on collision.
Next, make absolutely sure that you have both a front and rear light if you're going to light in any kind of low light conditions. Riding in the dark without a light is incredibly dangerous, because you'll be moving quickly, be hard to see, and be making very little noise. I like the silicone lights that don't require any mounting gear that you can put on and off your bike really easily (mine are "Snugg" brand and cost $15 for a pair on Amazon). These are great for riding around and being seen. However, they're not going to illuminate the road in front of you. If you plan on biking in really dark areas you'll want a bigger/brighter clip on light---I'd recommend the ones that are 1 watt or higher power output (most of the ones in the store will be 0.5 watts, which isn't ideal). Make sure you always remove your lights when locking your bike outdoors.
For locks, at the minimum you need a U-lock and cable lock.[1] The U-lock will lock your rear wheel and frame, the cable will lock your front wheel. Note that all of the cables you buy can be cut fairly easily (in a few minutes perhaps); the point of the cable is to deter someone from stealing the front wheel (which is fairly cheap), the U-lock is what will actually be securing your frame. I highly recommend the 5" Mini Kryptonite U-Lock. The 5" locks are not only the smallest ones, but they're also the most secure. U-locks can be easily broken by someone with a jack, if there's enough space to get the jack in between the bars of the lock to bend it. The 5" locks don't admit enough space for someone with a jack to get a hold on the lock. However, you'll really need an adequate rack to lock your bike with a 5" lock. For instance, it's generally not possible to lock your bike to a parking meter with a 5" lock whereas you can with a larger size. When you lock your bike, you need to place the U-lock so that it secures the rear wheel through the rear triangle of the bike. You generally should not directly lock the frame. By locking the rear wheel through the rear triangle, the U-lock is actually going through both the frame and the rear wheel (although it may not look like it!). The cable loops through the front-wheel and back around the U-lock.
In areas with high rates of bike theft, such as San Francisco, you'll need some way to secure your seat as well. I biked and locked my bike outdoors for years in Berkeley, Oakland, and Los Angeles and never had a problem with seat theft. As soon as I started biking in San Francisco, I got my seat stolen twice in the course of a month (both times having left the bike alone for less than an hour). So whether or not you need this really depends on where you live. Bike stores will sell special locks for seats. You can keep the lock on the seat all of the time because you'll only need to remove it in the rare situations when you need to adjust the seat height. If you don't like the look of a seat lock, or want to spend less money, you can also try securing the seat post bolt by using security bolts or hot gluing a BB into the bolt head.
If you're going to ride in the rain, I strongly recommend a detachable rear fender. Otherwise you're going to get a muddy butt. I've never found a front fender to be necessary; if it's rainy enough to need one, you're going to get drenched anyway.
[1] If you have security bolts for your front wheel, you can probably omit buying and carrying a cable lock.