I bought a trek 700 this weekend from someone for $140 so I can ride around with my son now that his training wheels are off

Front tire is balding. It's a hybrid bike

What kind of tires should I get. I've seen factory to continental tires. Is this something I can do myself?

You should be able to do it. There's plenty of you tube videos, I'm sure. Get a pair of tire levers to make it a lot easier and so you stand a better chance of not popping the tube.  Probably a good idea to replace the tube while you're doing it...save the old one for a spare just in case.

I'm not familiar with road tires, but Continental would be good. Kenda, WTB, Schwalbe, Panaracer, Vittoria, or any of the brands on the major online bike shops should all be fine.  I probably wouldn't buy a tire from Walmart.  You get what you pay for with tires, usually, so the cheapest may not be worth the trouble. Get the same size as is printed on the sidewall, to be safe.  You could probably go a little wider or narrower, but that could be a crapshoot, too.  

Or find and get to know your local bike shop.  They'd love to help you out, and they can be invaluable when you have an issue you don't have the right tool or the know-how to deal with.

I've owned a couple of 700 series Trek hybrids and both came with 700c x 38 tires.  They fit 700c x 40 mm tires, but wouldn't fit much bigger than that.  I would look for 700c x 35 or 700c x 38 as a replacement.  622mm BSD is exactly the same as 700c and some tires may use that nomenclature (ie, 622 x 35)

Tubes are all size specific but in a larger range (ie, a tube might be sized for a 700c x 35 to 700c x 45 tire).  

Just don't get a 26", 27" 27.5" or 650B tire.  None of those will fit your rims.