You haven't told us what kind of pedal system you are using - platforms, cages, "clipless", etc. As already pointed out, if you're using platforms, you probably aren't going to get to fast spinning so easily.

There's no magic cadence that's right for everybody, but as a general rule, beginners spend too much time turning harder gears slowly. I know I did. In a general way, you're already headed in the right direction - when you are going your cruising speed on a flat - whatever that speed happens to be - try going to the next larger cog and maintaining the same speed.

But don't stress about any of it. Keep riding and putting in the miles, making sure you enjoy them. The rest will come.