I'm not sure if I really "got into it", I've only done a handful of races spread over ~5 years.

I started off racing mountain bikes. Racing mtb gives you most of the skills you'll need for 'cross. Practice your mounts and dismounts and carrying the bike while running. You can practice these skills anywhere, but a nice soft grassy area is best when first learning the dismount. It helps if you increase your running-training in 'cross season too.

Check out local 'cross fliers. Most races have an A, B and C race. The C races will usually have beginners of all ability levels and you'll see more mountain bikes (since most of these folks haven't sprung for a full on 'cross bike yet).