So far, I have considered only three algorithms, namely, minimax, alpha-beta pruning, and Monte Carlo tree search (MCTS). Apparently, both the alpha-beta pruning and MCTS are extensions of the basic minimax algorithm.
Given this context, I would recommend starting out with Minimax. Of the three algorithms, Minimax is the easiest to understand.
Alpha-Beta, as others have mentioned in other answers, is a strict improvement on top of Minimax. Minimax is basically a part of the Alpha-Beta implementation, and a good understanding of Alpha-Beta requires starting out with a good understanding of Minimax anyway. If you happen to have time left after understanding and implementing Minimax, I'd recommend moving on to Alpha-Beta afterwards and building that on top of Minimax. Starting out with Alpha-Beta if you do not yet understand Minimax doesn't really make sense.
Monte-Carlo Tree Search is probably a bit more advanced and more complicated to really, deeply understand. In the past decade or so, MCTS really has been growing to be much more popular than the other two, so from that point of view understanding MCTS may be more "useful".
The connection between Minimax and MCTS is less direct/obvious than the connection between Minimax and Alpha-Beta, but there still is a connection at least on a conceptual level. I'd argue that having a good understanding of Minimax first is still beneficial before diving into MCTS; in particular, understanding Minimax and its flaws/weak points can provide useful context / help you understand why MCTS became "necessary" / popular.
To conclude, in my opinion:
- Alpha-Beta is strictly better than Minimax, but also strongly related / built on top of Minimax; so, start with Minimax, go for Alpha-Beta afterwards if time permits
- MCTS has different strengths/weaknesses, is often better than Alpha-Beta in "modern" problems (but not always), a good understanding of Minimax will likely be beneficial before starting to dive into MCTS