Climbing

Well at your level you just need to be learning movements naturally. I.e. keep going climbing. Climb as often as you can while still recovering properly between sessions. My advice for learning basic technique is to watch better climbers climb, ask them to show you how THEY do a boulder that's at the top end of your ability, and try to mimic them. Work on making your V2s feel effortless. Don't just move on and forget a boulder after you top it the first time.

When you DO finish a V3 you're going to be sitting at that grade for probably a lot longer than you sat at V2s, and longer still when you're at V4, so... uh... get used to that is my advice. How quickly you can progress depends a lot on your body composition coming into the sport as well as how often you train (with proper recovery). Stop focusing so much on reaching the next grade, start practicing the basics, and the grades will come naturally. You MIGHT be able to get away with sending the odd juggy V3, but you're never getting anywhere without basic technique, which only comes from a lots of practice.

Edit -- by the way. Usually when people talk about plateauing at a grade... they mean they've been stuck there for years, not weeks. Yes, beginner gains are much faster but a few weeks is nothing, your body is just starting to think about maybe making muscle and tendon adaptations for climbing... with the all-important tendons being much slower to adapt than muscles.

I'd say at V3 is where I had to really become aware of balance and where my center of gravity was relative to the hold I needed to reach next. Look for little adjustments you can make to make your way up the wall easier, see how it feels to twist your hips one way versus another, feel the difference in having a foot higher as opposed to lower.

Climbing more with all that in mind will be important as you'll gain an instinctive understanding of how to move your body on the wall.

V3 and v4 are typically where beginners find themselves struggling.

What you may find helpful is watching skilled climbers on YouTube or in your gym. I like Magnus and Bouldering Bobats on YouTube. Seeing how skilled climbers approach problems can be helpful for learning techniques.

If your gym offers classes, sign up! It's a great way to learn with a group and improve directly.

Ultimately, it took me a few months of climbing before I was easily doing V3's.

Your muscles are going to get stronger way faster than your joints. A lot of holds that are ~V3 and up (depending on how your gym grades things) will end up putting a lot more strain on your finger joints/ligaments than the big, juggy kind of holds you see at the V0-V2 range. If you're having trouble with really crimpy holds, that's probably a good thing if you've still only been climbing for a couple of weeks. You really don't want to sprain a pulley tendon, it sucks.

So, for now, be very careful about putting excess stress/strain on your joints, especially your finger joints and elbow joints. Those tends to be the easiest to mess up, in my experience. I would advise against trying to do anything excessively strenuous until ~1/2 a year of climbing consistently (2-3 times a week). Until then, focus on climbing while trying to keep your center of mass as stable as possible, particularly when moving from one hold to another. Very few typical V0-V2 routes require a lot of dynamic movement to reach the next hold if the proper technique is applied to the move.

Another thing to focus on is reading your route and executing it. If you're physically fit, there's a tendency to just muscle through the lower graded routes since you can without applying the techniques to do the route without using the techniques that the setter was putting in the intended beta. If you're stuck on the V3 plateau (which is reaaally common, so don't get discouraged!), try doing the routes you can do using as little effort/strength as you can and try to force yourself to use those intended techniques. It will help you become more aware of the toolset available to you in particular situations and work out the muscles necessary to execute those techniques that are actually required for being able to send more difficult, complex routes.

It's very common for people who get into climbing or bouldering to get hooked and obsess over getting better.

For at least the first few months, your body is going to get used to the strain that you put on it. The easier grades in the gyms will all typically be much easier on the joints and soft tissues (ligaments and tendons) which take a long time to strengthen compared to muscles. Crimping in particular is something to be careful with. Look up photos for the difference between a half crimp and a full crimp. Avoid full crimps as much as possible because they are very strenuous and an easy way to get injured, especially as a beginner.

That said, more than specific training, what will help you progress is practice. Climbing is a skill-based sport, and is a lot more about positioning and balance than it is about strength.

A few things to keep in mind:

First things first, learn how to fall safely. Never put your hands down while trying to catch yourself, and try to roll onto your back when you fall. Letting yourself crumple over onto your back is much safer than trying to land standing up on your feet. You're a lot less likely to roll an ankle or strain your knees, and will save future you from back and joint pain.
Keep your feet on the wall, and push with your legs instead of pulling with your arms.
Use your legs and hips to adjust your center of balance to be able to move better and avoid "barn-dooring". Look up flagging, for example.
Avoid over-gripping. As a beginner, you do this for sure. You only want to grip the holds as lightly as possible so that you're not falling off.
Keeping your hips closer to the wall helps keep your weight on your feet and allows you to reach higher.
Use your toes on the holds, not the middle of your foot.

Last of all, don't worry too much about chasing grades, and focus on having fun.