One tip which I read somewhere else on the forum, and which I’ve found useful, is this:
(1) Match 5s first
(2) Match 4s from the top down
(3) Match 3s from the bottom up
I also try to think about whether, all things being equal, I should match the lower value or higher value gem. Lower value gems make more matches; higher value gems allow you to push chests down the board or clear space.