Here's my condensed bike maintenance checklist. This is by no means comprehensive, but I'll try to give you a rough idea.
1. Drive train
First the simple stuff. Lube the chain regularly. (Like, every few days if you ride daily.) Turn the bike upside-down and oil the whole chain, then shift through all the gears so they get oiled too. Then take a rag and wipe excess oil off the outside of the chain by holding the rag on the chain while turning the crank. You want oil in the rivets in the chain; the stuff on the outside does nothing but attract dust.
Every so often, take your chain off, clean with solvent, dry, and then put back on and re-lube. This is simple and totally worth learning to do yourself. You will need a chain breaker, which you can find at your local bike co-op, or you can buy your own for <$10. While you have the chain off, you can clean the crud off your gears and rear derailleur (the thing that moves the chain from gear to gear) with a stiff brush or rag. Old socks are good for running between the gears. Ideally, if you're riding daily, this should be done every one to two weeks (note: in reality, I'm too lazy to do it that often, as are most bike commuters I know, but I'm talking theory here).
Cleaning your drive train regularly will probably solve some of the shifting issues, plus, the more often you clean it, the slower your gears will wear down - the oily, pasty crud that forms on gears acts like sandpaper and wears your gears down faster.
2. Shifters/derailleurs
Check the tension on your derailleur (shifter) cables, and check that you can shift through all the gears. There are a lot of potential reasons why your bike might not be shifting smoothly, too many to really get into here. I can virtually guarantee, though, that if you haven't maintained your bike in a while, your shifter cables need to be tightened, as they stretch out over time. Tightening the shifter cables requires only a wrench and a pair of pliers, and you can easily learn to do it yourself.
3. Brakes
Check the tension on your brakes regularly. The brake cables also tend to stretch out over time, making braking less efficient. Make sure you can stop quickly. You should be able to brake with full force without the brake levers touching the handlebars. Make sure your brake pads are aligned with the wheel rims (when you brake, they should be in full contact with the wheel rim, not hanging over the edge, or touching the tire). Make sure the brake pads don't rub on your wheels when you ride. Check your brake pads for wear, and debris. Brake pads need to be replaced once in a while, just like on a car.
4. Wheels
Check that wheels are in true. Turn the bike upside down and spin the wheels. Do an eyeball check - are there any obvious wobbles? (You might also be able to see these wobbles while you're riding.) Are there any spots where the wheel rubs on the brake pads? Feel the spoke tension by grabbing the spokes, two neighboring spokes at a time, and squeezing them together. Are any of them noticeably loose? If you notice any of the above things, your wheels need truing. (Another complicated topic that I won't get into here, but your friendly mechanic or bike co-op staff can help you.)
5. The bearings inside your wheel hubs, headset (where the handlebars attach to the frame) and bottom bracket (where the pedals turn) need to be taken out, cleaned, and re-packed every so often. I've heard a lot of different opinions on how often this should be done, but I'd say minimum once a year for the wheels and bottom bracket, ideally once every three to six months. To be honest, I almost never touch my headset, because it's much less exposed to water and dirt than other moving parts, but some people will tell you it should be cleaned regularly too. If you take your bike to a mechanic for an annual tune-up, they will sometimes re-pack your wheel hubs and bottom bracket as part of the tune-up package. Ask.
6. Tires
Change when the side walls dry out and crack, or when the treads wear out, or when you get a large puncture (e.g., from running over a screw or a big piece of glass). Keep a tube patching kit and a couple of spare inner tubes around the house for fixing flats. Ask the staff at the bike shop to help you get the correct sizes of tires and tubes - it's not always obvious.
I always say, if you own only one bike tool, it should be a pair of plastic tire levers, for taking off and putting on tires. You can get these for a buck or two at the bike store. (A lot of people use screwdrivers for changing tires, causing a lot of unnecessary flats.)
There are a lot of good bike maintenance books out there. I can recommend these:
http://www.amazon.com/Big-Blue-Book-Bicycle-Repair/dp/0976553007
http://www.amazon.com/s/ref=nb_sb_ss_i_1_45?url=search-alias%3Dstripbooks&field-keywords=zinn+and+the+art+of+mountain+bike+maintenance&sprefix=zinn+and+the+art+of+mountain+bike+maintenance (this is more unorthodox and personal - a great DIY book)
There is a lot of information about bike maintenance to be had on the internet.
If you are good with cars, you should be able to learn to do any of the things I mentioned above without too much trouble, assuming you have the time and interest. If you want to learn more about your bike, I highly recommend getting involved with your local bike co-op. They often offer help with repairs and maintenance, and courses on bike mechanics, and they have tools you can use, including a lot of bike-specific tools. Even if you don't plan on being a DIY mechanic, it's worth it to learn some bike basics and terminology. I'm unfortunately not very familiar with the local resources for cyclists in Seattle, not being from there, but here's one place I know of:`http://www.thebikery.org/
There you go. That's kind of a long-winded answer, but I hope it helps.