What's the best approach to drawing a colored picture
I also don't know if I should do a basic sketch in pencil, then erase
 it and go over it in colored pencil

You absolutely can do this - try using a harder pencil like a 7H or 5H and keep your lines light so you won't have to cover up as much later.

What is the best technique?

The question is too broad to answer well, but I think your first instinct to try colored pencils is a good one - they are forgiving and familiar.

Are there special art colored pencils that would be significantly better than regular crayolas?

Yes, there is a big difference between Crayola and the higher end colored pencils by Prismacolor, Derwent, etc. At many art supply and craft stores you can buy individual colored pencils if you'd like to see the difference without dropping a lot of money on a full set.