The specifics of this are unique to the person, but in general: Just keep going climbing.
There are (very broadly speaking) two factors which determine how hard you can climb.
- Technique, or how you move and position your body
- Strength, or how strong your body is
Going climbing will improve both of these things. You'll get comfortable in the strange positions climbing sometimes puts you in and your muscles will get used to gripping holds of a variety of shapes. Essentially, practice makes perfect, or at least better.
Now, there are ways to improve each of these things on their own (footwork drills, hangboarding, etc.) but these are a shockingly bad idea for the V0-V2 climber.
It is unlikely, as a new climber, that your muscles and tendons will respond well to targeted workouts; it is likely, as a new climber, that hangboarding will injure you. And while technique exercises won't hurt you... they're kinda boring! I suggest enjoying regular routes, and asking fellow climbers for technique advice when interested.
The advice above is pretty general, but I'm guessing you're also feeling a bit of...
The Plateau/The Jump
This is when you feel that your rate of improvement has decreased (plateau) or the change in difficulty from one level to the next is insurmountable (jump). Perhaps when you started climbing, a V0 was doable and a V1 was very hard, but you could at least attempt it. Now you feel that V1s are doable but a V2 is impossible. This is very common, and V2 won't be the first time you experience this. Here are a few tips for getting past it.
- Climb more. Go twice a week instead of once. Stay at the gym for 3hrs instead of 2.
- Work on the parts of harder climbs that you can do. There is often a big difference in style of climb between V0-V2, and getting used to the new type of movement will help you improve.
- Work on individual moves at your limit. Find a few moves that look cool to you and work on them. Try them every different way you can think of. The feeling of sticking a move after trying it over and over for a week is incomparable.
- (optional) Ask for advice. When you're working on your one hard move, talk to the local crushers and see if they have a suggestion--it's amazing how much a well placed backstep or dropknee can change how you feel on the wall, and sometimes new climbers won't recognize (or even know) how to identify the opportunities.
Above all, go out and have fun.
Edit after edit to question:
Most of my answer involves climbing more, and you say you can't get to the climbing gym more than twice a month. I hate to say it, but you're going to have a hard time improving significantly. As with most technical sports, you can't improve without actually doing it. Sure, you can do grip workouts and yoga and situps and more, but in all likelihood you'll get on your next V2 and it will feel just as hard as before.
An analogy, in reverse: I recently started going to a regular gym. I can climb pretty hard routes on tiny crimps with huge jumps. But when I go to the gym I can only do 5 or 10 pull ups--and what gets tired first? My grip.