Turning left or right going forwards involves cross-overs. This means that as you turn, you cross the outside leg over your planted leg. At all times, you're pushing with the edges of your ice skates. It's the combination of crossing over your legs and the pushing that produce speed in a turn.
Cross overs were the first ice skating technique that I really figured out, and when you learn to cross-over, it should feel as if you've solved a fundamental puzzle to ice skating. Remember those "Magic Eye" puzzles? If you remember your feeling after "seeing" the magic images for the first time, learning cross overs is really like that. For a sporting analogy, it's similar to your first pure strike of a golf ball.
Most of the players I met were okay with cross-over turns in one direction, but the other direction brought a certain hesitation. Strive to learn your cross-overs in both directions. Each team in an ice hockey game will typically skate in each direction, so you'll need to get used to cross over from your "weak side."