I've done a bit of work on old, neglected bikes at our local community bike shop. It's rewarding to bring a forgotten bike back to life. There are usually a lot of things that can be improved, so it's important that you prioritize carefully. Fix the most important things, get riding, and then fix the rest as you go along.
You haven't told us much about your specific bike, so I'm going to write generically.
The most important thing to do when reviving an old bike is take care of the bearings.
Normally bikes have bearings in 5 places:
- front hub
- rear hub
- bottom bracket
If bearings are too tight or too loose, they will will crack and chip. The rough edges will cut in to the races, and you'll have to replace them. In the case of wheels, that can mean a new hub, which can mean a new wheel.
The freewheel is difficult to check, and less likely to need immediate care, but the other 4 deserve immediate attention. Check that they turn easily, without much resistance and without a "gritty" or "notched" feeling. Headsets are particularly susceptible to notching at dead center. Also check for play in the bearings. If you try to rotate against the axis of a bearing, there shouldn't be any play, or maybe a very tiny amount.
If the bike has been neglected, especially if it was left out in the rain or ridden in the rain and not serviced, you should overhaul these bearings. Basically, you open them up, throw out the old bearings, clean the races (the paths the bearings roll in), fill with fresh, clean grease, insert new bearings, reassemble, and adjust to be "just tight enough". While you're in there, it's a good idea to examine the races to ensure they are smooth.
Pop out the brake and shifter cables from the control levers. Slide the housings back and forth over the cable. Look for rust, frayed wire, and sharp kinks. Any are signs that you should replace the cable, but if the wear is light, you can get away with deferring that replacement. For now, drop some graphite- or teflon-based lube in to the housings, and slide them back and forth to spread it out. Avoid getting lube on the rims, where the brakes need to grab well.
The brakes are the most important piece of safety equipment. Ideally the brake pads hover close to the rims, but don't rub. Pads are clean, as are rims. Brake action is smooth and easy. Both pads contact the rim at the same time, and are centered on the rim's braking surface. They especially don't hang off the inside (where they could pop off the edge and jam) or the outside (where they will rub the tire and cause a blowout). When you release the brake levers, the brakes quickly snap to open.
You'll need the wheels true before you can reach brake Nirvana. (See below.)
For now, just make sure that both brakes work - that you can stop in a hurry with either one.
If the bike has been neglected, your chain is probably rusty, and will need to be replaced. If it's ridable, then clean and lubricate it carefully.
Examine the wear on the chain and the front and rear gears. As chains age and stretch, they place force unevenly on the gears, which causes them to wear to match. If you then put a new chain on old gears, the chain will wear out before its time. Similarly, new gears with an old chain will wear fast.
If the chain is mildly worn and the gears are healthy, replace the chain now. If the chain and gears are both very worn, you can keep riding, but plan to replace both soon. They may slip under heavy loads, too.
An easy way to check the health of a chain is the Park Tool Chain Wear Indicator CC-3. Keep one of these on hand so you can check your chain regularly.
Check that the gears shift smoothly under load, and the chain doesn't try to come off the gears when you shift to the shortest and tallest gears. You'll need healthy shifter cables before you adjust these.
When wheels are neglected or abused, they will go out of true, go out of round; the rims will get bent or dented; the spoke tension will be uneven, making the wheel weaker.
The first thing to do is evaluate the state of the wheel. You'll need well-adjusted hubs to do this. Spin the wheel and watch the gap between the rim and the brake pad closely. If the gap stays pretty constant, you're in luck. If it wiggles, you have work to do. Spin the wheel again, and let a screwdriver or your fingernail bump in to each spoke. Listen to the sound. If the pitch jumps around, spoke tension is uneven. Listen to good, new, hand-built wheels to teach your ears what to listen for.
Even if you're not going to true your wheels yourself, it's a good idea to lubricate the nipples now. Give the oil some time to work in now, so a mechanic has an easier time with truing later. Put a drop of oil in the crevice between the nipple and the spoke.
Also check your tires. Look for cracking in the rubber, wear in the sidewall, and depth of tread. If you can see threads anywhere, you're close to a blowout.
Pump the tires up to somewhere in their rated range. Personally I like the lower end of the range, but each person is different.
Stand over the front wheel and grab it tightly with your legs. Try to turn the handlebar. It shouldn't budge. Try to rotate the handlebars in the stem; it also shouldn't budge.
Pull the seat tube out and make sure there's some grease on it, so it doesn't try to fuse in there.
Take the wheels off and put them back on, and make sure you know how to use quick release properly.