Sounds like a loose headset to me.
Put the bike on flat dry level ground, hold the front brake firmly, and push forward on the handlebar.
Does it rock a tiny bit?
It should be absolutely still, (or, if you push hard enough, the rear wheel should lift off the ground).
If it rocks slightly, see if its moving where the stem goes into the headtube
There are two kinds of headset.
One looks like this:
If it looks like that, loosen the upper lock nut, tighten the top race just until its snug, then tighten the lock nut against it.
The handlebar should still turn smooth and easy, but there should be no lateral play. Try the thing with the brake again, and it should be perfectly still. It may take several trys to get it perfect, not too tight not too loose.
If it doesn't look like the above image, it will look like this:
In which case you loosen the pinch bolts, then snug the bolt in the top cap. Again, tight enough that there is no play, loose enough that it turns without binding. When its perfect, tighten the pinch bolts again.
If the headset isn't loose, my second guess would be the axle.
however, do NOT just tighten the outside nut. There are two nuts in a row, and if you tighten the outside one you'll just make it extremely hard to get off, or possibly damage it.
You need to take the wheel completely off, and then loosen the outside lock nuts.
The cones (the threaded part that pushes against the bearing) is the part that needs to be adjusted to take out axle play NOT the nuts. The cones require a very thin wrench (called, not surprisingly, a cone wrench).
You can not adjust a bike wheel without a cone wrench. Don't even try. They aren't terribly expensive.
(Also, for future reference, you can not adjust a rear wheel without taking off the gear set / cassette / freewheel, for which you need a different special tool)
The process is basically the same as for the headset - make it as tight as possible without it binding or dragging or having any extra resistance, which is best done by trial and error. Make sure to hold the cone in place while tightening the locknut, or you will move its position and have to start over.
Yeah, bikeshops charge a lot. The ones I work at charge a lot more than I used to charge, however, my old shop was funded by grant money, and they have full time staff to pay, plus rent, electric bills, stuff I never used to have to deal with. Also, a good bike mechanic has just as much specialized knowledge as a car mechanic. I never realized how much I didn't know about the finer points until I started working with someone who does know it.
And you really never know with customers...
I once had a customer insist I was responsible for her flat tire because I had previously adjusted her gears.
Recently I mentioned to a different customer that her crank was loose (she had come in for something else) and then when it came off days later, she told the shop she brought it to that I had just replaced it! I had just said it needed to be done, not that I had done it. And don't get me started on the people who ride department store bikes...
