Your description sounds like the wheel is out of true, out of round, or some of the spokes are so loose that they're flexing as the wheel rotates. However, for it to be as severe as you describe I would expect it to be visible on spinning the wheel and watching it, even to a novice mechanic.
Whatever the case, you should take your bike to a qualified mechanic. The problem may be fixable, possibly as simple as tightening and loosening the appropriate spokes. This is a relatively difficult job for a novice mechanic though. It is also perfectly possible that you need a new wheel. An experienced mechanic would have to visibly inspect the wheel to be sure.
If it's a simple fix, it could be as low as $15 US. If you need a new wheel, I would expect it to be around $75-125 for a bike that age depending on what quality you would like to get.
I would also recommend checking with a couple mechanics. Some are quick to sell new parts when an older part is serviceable.