Actually, the rules for the buyer are the same, whether buying a handgun from a dealer or from an individual. Rules for both dealer and individual sellers are ESSENTIALLY the same too (there is no separate standard or statute for each), except that an individual doesn't have the practical means of verifying the legal status of the buyer as does a dealer. An individual can, though, and should (my opinion) ask to see legal proof of residence of the buyer and whatever else they feel satisfies them that the sale is legal.
However, in Texas (thank goodness) no registration or other paperwork is required if you buy a gun from your brother, next door neighbor, or total stranger. The FBI check and "yellow form" is a requirement of the dealer only. That will remain so only if we all continue to vote and support our good legislators with $$$$ to help them stay in office.
Straw purchases between individuals are also illegal, just as they are from a dealer.
You must have reached your 21st birthday to purchase a handgun in Texas (or in any other state, to the best of my knowledge).
Re the comment about Colorado--wasn't the individual prosecuted who sold a handgun to one of the school shooters?