Correct, don't dryfire the rimfire.
What aspects of shooting are you wanting to work on, in particular? If you're just starting out, then there are some things you need to develop, and some things that will only come with practice. From what I'm reading, you don't have the time or resources right now to get a lot of live practice in, so you'd better work on fundamentals.
First, the grip. Watch this:
Shooting is like golf in a lot of ways, the first being that the grip is INCREDIBLY important. The second is that it feels weird at first, but will become second nature.
After establishing the correct position from that video, the next big thing to learn is that it is your weak hand, not your strong hand, that supplies most of the gripping force. The biggest detriment to accuracy is what happens when someone pulls a trigger. Muscles move, recoil anticipation comes in, flinches start, etc. The first thing to eliminate is muscle tension caused by gripping the gun too hard with the strong hand. You want a firm grip, but only up to the point that you can still calmly, smoothly work the trigger without any of the other hand muscles sympathetically twitching. Think of the gun + strong hand as the entire tool, and the weak hand as the support that moves the tool around.
Last tip for today, develop a good stance; this means be relaxed, don't lock any joints, and lean forward. You want your weight 70/30 on the front of your feet, to soak up recoil. (note that some of these tips will not be as relevant until you move up to centerfire pistols.)
There are a few more fundamentals, but that's enough for now. Let me know if you have any particular questions.