How can I balance the tires by myself?
Static balancing (balancing without a machine) isn't hard, but it can be a little time-consuming. Static balancing is good enough for professional roadracers, so it will work just fine for you. If you feel better having your wheels dynamically balanced by a machine, you can take it to a shop and pay for that peace of mind. The primary advantage of balancing machines is that they're faster. Static balancing can take 5-20 minutes per tire to do correctly.
See also: Tire Mounting Hints & Tips
Before you mount the tire
The first step to balancing the tire is to mount it in the proper position. To do that, you need to find the wheel's heavy point before you mount it. Remove any old balancing weights and clean that area, using a solvent or Goo-Gone to get off any glue that held them on. Set the wheel up between two level wheel holders. Automotive jack stands work well for this purpose, as do two milk crates. Keep them level and parallel and far enough apart that one end of the axle can rest on each edge without the tire or disk(s) rubbing on the setup. Rotate the wheel 90-120 degrees (without the tire on) and let it go. The heaviest part will rotate to the bottom. If there isn't a heavy spot, then it won't rotate from where you let go.
Do the test a couple times, and also rotate both directions. Mark the low spots with easily-removable tape. Then, average those together for the actual heavy spot. You'll only need to do this once for the lifetime of the wheel. Heavy spots don't migrate. Mark the spot on the inside of the wheel with a scribe, which will leave a permanent mark but won't damage the aluminum. Just don't scribe in the part of the wheel where the tire bead sits, as it may cause air leakage.
When you go to mount the tire, there should be a paint spot on one of the sidewalls or inside the tire. This is the lightest point on the tire, as determined at the factory. This point needs to be mounted at the wheel's heaviest point. You've just found your wheel's heavy point, so put the tire's paint mark there. This should make the balance pretty close to begin with.
Note: Most shops will align the paint mark with the valve stem, as that is often the heaviest point, but not always. That's why you just checked. Often it's quite a ways from the stem, or even completely opposite. As you can see here, the valve stem is off to the right, and very much not the heavy spot.
Before you start, you'll need to buy some weights. There are 2 main kinds of weights: clamp-on and stick-on.
Stick-ons are cheap, easy to come by, and do the job well. You can get these from JC Whitney, eBay, and most automotive stores. These usually come in a strip of 7-gram (1/4 oz) weights that break apart like a Hershey bar.
Place your jackstands or other straight-edges level and parallel as before. Clean the axle, insert it into the wheel, and set the combination on the straight-edges. You will get better results if the axle is free to rotate on the surfaces of the straightedge, since there may be some stiction in the wheel bearings that might prevent smooth rotation of the assembly if the axle is constrained.
As you did when finding the heavy spot, hold the wheel up 90-120 degrees and let go. Don't spin the wheel; that'll just take more time to slow down. An unbalanced wheel will behave like a slow-motion pendulum and come to rest with the heaviest side at the bottom. When it stops, note or mark the position (chalk on the tire works well). Do this repeatedly, turning the wheel in both directions. A balanced wheel should not stop in any one position. If it stops in random places each time, the tire is balanced. If it repeatedly stops at the same place each time, a wheel weight needs to be affixed.
Put your weights as close to the centerline of the wheel as possible.
Since the wheel settles with the heaviest point at the bottom, you need to put the weights on the top, at the lightest point. Tape the weights to the right spot on the rim using tape that is easily removed. These are temporary for now.
Keep turning the wheel in both directions and adding or subtracting weights until the wheel doesn't stop in any one position. Be patient for best results.
Once the wheel doesn't stop at the same point every time, mark the position of the temporary weights on the tire with chalk and remove them. Clean the wheel where the new weights will go.
If you have used several small pieces of weight to get the right amount, line the temporary pieces up and cut off a single piece (may take more than one) of lead to be the permanent weight. Press it into place so its shape conforms to the curve of the wheel. Then attach it to the rim. The weights should have their own adhesive on the back. If not, use double-sided foam tape. If you want to ensure that the weights will stay on, you can add a bead of hot glue around them.
Re-check the balance by letting the wheel fall from different positions, both forward and backwards. Then, regrease the wheel bearings and axle and reassemble.