The youngster has proven that he or she actually knows enough of the basics to use a telescope. First and most importantly do not, I repeat do not, buy a cheap telescope that can be found in a discount store. The child will be one hundred times better off with a good pair of binoculars than he will be with a cheap telescope. Secondly, get thee to a library. You and your child need to read everything possible about telescopes.
Many books offer good information about different kinds of telescopes.
There are many different types and many will fit the needs of a first time owner. Both of you will need to decide which one will best fit the needs and wants of the young person. Find the local astronomy club and go to a few meetings. Go to a few star parties. Talk to people who have spent money on good telescopes.