Federal law states that you may not purchase a handgun if you are under 21. Makes no difference if it is from a dealer or from a private individual. State law does not change this. You may POSSESS a handgun if you are 18 or older. It must have been given to you by someone that did not require a transfer from a licensed dealer or a private individual (ie, legitimate gift).
There is no "registration" of firearms in this country (yet

) so you don't have to transfer your name to the gun. If you purchase from a dealer, you must complete an ATF form 4473 but from an individual there is no paperwork at all unless the seller wants your name and address for his own records.
If you legally acquire the gun, then of course you can legally take it to a range and shoot it. Know your state and local laws regarding transporting a handgun also. Hope this helps you.