The care and feeding of your bike

You've found just the kind of bike you need, tested the fit, inspected it, ridden it, and found it to be fun. It's time to buy your bike and bring it home.

There were no serious problems with the bike during your thorough test drive, but you did find that the rear tire has brittle, cracked sidewalls and the chain will probably need to be replaced within a year due to corrosion. The seller did not describe these issues during your correspondence, so you bring them up and make a counteroffer $20 lower than the asking price. The seller agrees to take off $15, and you decide that it's a price you can live with. $15 will cover more than half the cost of a tire and you feel like replacing a tire is a DIY project you can handle. You'll deal with the chain when the time comes. You pay her the cash and take home your new bike!

And that's the way to feed your budget bike: replace things when they need replacing or you want a project, lube the chain when it gets noisy or a bit rusty, clean things as needed. There are many guides around about bike maintenance, including some video tutorials on youtube.

Tools and Toys

In addition to a bottle of bike lube (i recommend Boeshield T-9, but remove the sticky, nasty oil on the chain first), you'll want to acquire a few accessories to make your bike owning experience safer, more enjoyable and reliable. Here is a partial list, but don't let it stop you from buying your bike a cool new toy not on the list if you feel so inclined.

A helmet is a useful, disposable piece of safety equipment that absorbs impacts instead of your head. It should be replaced any time it is involved in a collision. Inexpensive helmets certified for sale in the USA are no less safe than expensive helmets because, after they are certified to meet certain safety requirements, the manufacturers spend their resources making the helmet lighter and more ventilated rather than safer.
A lways lock your bike! I left my bike outside the door of a bank in Miami while ducking inside the vestibule to quickly use the ATM, keeping an eye on the bike. In an instant, the female half of a tattooed couple tried to ride off on it. She was foiled in the attempt because it was too tall for her and in a high gear (my concession to security in this situation), and they ran off as i rushed out.
There's nothing that will stop a determined thief, but a good bar ('U') lock will deter most would-be criminals. I recommend a very compact version to make it easier to carry. I currently use a compact u-lock in combination with a length of plastic-sleeved "airline cable" (I've only found it at REI). The airline cable has loops woven into each end. Depending on the situation and bike, i use it to secure everything from the saddle, front tire, and helmet when i'm being cautious to the opposite extreme of just locking the rear wheel to the frame with the bar lock and looping the aircraft cable around a pole. One of the nice things about this combination is that you can get some security for the bike even when there's nothing your lock can fit around.
Being cautious about where, when and for how long you leave your bike are more important than how it is locked, but always lock it when it will leave your sight unless you don't mind losing it. On a side note, that's a viable strategy if you don't want the hassle of worrying about it being stolen: buy a junker bike and just replace it if it walks off. I left my $20 police auction department store bike unlocked for several weeks near the end of the school year before i finally managed to get rid of it so i could get a better one. ;)
Patch kit and spare tube
Replacement bike tubes are about $7 (and it's worth having a spare if you rely on your bike). Make sure you get the correct size for your tire. Even though they are rated for a range of tire widths, they are not entirely one-size-fits all, even when you match the circumference (26" vs 700c) as discussed in the section on tires. I've known people who just replace the tube when it gets punctured, but not only is this wasteful, but patch kits can be had for as little as $1 if you buy in advance, and don't require you to remove the entire tube from the bike to patch it. That's only a fraction of the cost of a tube, making a patch kit the best value for money in a bike accessory! Make sure you buy the kind that comes with a tube of rubber cement and patches, because the "self-adhesive" ones are useless. I've bought them on several occasions to try them out, and they are always laughably inadequate. The most important tip when using the old fashioned kind i'm recommending here is to let the glue dry for five minutes before putting on the patch. It's counterintuitive and is probably the most common mistake people make. Incidentally, both of these accessories are much more useful when you also have a pump...
A common source of frustration for would-be bikers is keeping a tire adequately pressurized. An underinflated tire is a lot more work to ride on, and is easily punctured. Having a good pump will go a long way to improving your biking experience. If you are using the bike for transportation, you will sooner or later regret not having a portable pump. If you have several bike riders in your household, you will definitely want to get a good floor pump. I recommend portable pumps by Lezyne (get one appropriate for your tire type) and the Topeak Joe Blow Sport pump for a sturdy household model.
Mine has tire levers, allen (hex) wrenches, and a chain tool. If you have anything but big, fat tires on your bike, you'll want at least tire levers to make repairing flats easier.
If you intend to ride at dawn, dusk, or night, you will need lights. Some people even use bright blinky lights during the day to improve their visibility to drivers. Don't spend a lot of money on a fancy bike headlight, as there are better options (that's a post for another day). The Planet Bike Superflash is about as close to a de facto standard for rear blinky light as could be, but it does tend to fall apart and get lost. There are some less expensive knock-offs of that light that are about as good. I'll be posting about some other options.
Cargo rack, handlebar bag, panniers, trunk bag, trailer, etc.
If the time comes that you need to carry cargo on your bike, there are a lot of ingenious and effective options. For a rear cargo rack, the Topeak Explorer Dx line is great, but there are some details to consider, and that's a post for another day.

Well, you've traveled the road, and you've ended up with a bike. Congratulations!