The great book New Rules for Classic Games by R. Wayne Schmittberger has some suggestions for different ways to play Trivial Pursuit. I think the best one is to add challenges. If you think another player has answered incorrectly, you challenge them and give your own answer. If you're right, you get a pie for the category you answered. Already have that pie? choose your colour. If you already have six pies, he suggests removing a pie from the opponent, but I think you might rather call that a winning condition. If your challenge is incorrect, he suggests losing a pie, but that will slow down the game. I suggest you just skip your next turn.
Another option you might like is to play without the board. One suggestion in the book is called Going Out. Each player takes a stack of cards, maybe 24 high. On your turn, choose a category to start with, read the question aloud and answer it. Flip the card to check and discard it. If you were correct, answer the next category of the next card. If you were incorrect, it's the next player's turn. First player to discard their last card wins. If you want to be extra fair, give remaining players one more turn until you get back to the first player. You may need to have some kind of trivia shoot out to break ties.
There are a bunch more options in the book, including Trivia Tic-Tac-Toe. It looks like it's out of print, but the Amazon page includes the "look inside" feature, and I just searched for "Trivial Pursuit" to find all these rules. I had to sign in first, your mileage may vary.