The answer itself is this: on the ice you create rotation the same way you do on normal ground - by spinning yourself (by creating angular momentum).
Ever heard of "an object in motion stays in motion?" That applies to angular momentum (rotation), too. It means that once you've created rotation you'll keep spinning - until you create rotation in the opposite direction to stop it.
Everything in higher-level skating revolves around these physics, around creating, killing and handling momentum. That's how you skate in a curve - you just skate forward while possessing a controlled amount of angular momentum. The ability to to easily skate in curves and to control their radius is called edgework.
How to create and manipulate this momentum? Free skating techniques, in essence, let you use your arms, shoulders, free leg, hips and torso rotation for this exact purpose by moving them in particular ways. It's a whole-body, whole-mind affair that is very complex and, in my experience, takes more than ten years to master. Not even all championship skaters have mastered it (jumps are more important for winning).
Now, I assume that what you want is not a curvy series of rockers and counters, but a simple curve, or, say, a little three-turn. Know how those trainers always tell you to hold your arms up? The reason for that is not aesthetic. Turn your outstretched arms left, complete with your shoulders, like you're a windmill facing up, and that'll create rotation left. Same for right. If you're wondering why high-level skaters don't hold their arms up all the time for things like three-turns: Their movements are much more efficient, so they can turn themselves using only their shoulders in an almost unnoticeable way.