The second that your wheels break ground your plane has no idea what's happening in relation to the ground. Flying is 100% about how air is moving over the wings. So yes, a plane can be in flight and be motionless or even moving backwards in relation to the ground. I've had several takeoffs with my Cub where a gust of wind lifted it 2-3 feet off the ground before I had even lifted the tail.

That said, with wind also comes turbulence, and if you are only flying at the speed of the wind, the turbulence is going to toss your plane around a lot. On a "gusty" day the turbulence can actually be more powerful than the wind itself, not to mention that gusty wind starts and stops. So if you are putting along in your trainer a gust of wind will hit the nose and raise it, which reduces your ground speed and puts the wing in a position to stall. Then a lull follows which puts the wing into a stall even though your engine is still running and the plane was flying fine a second ago. You have to work the elevator to keep the nose level or even down a bit when the gusts hit and keep more throttle in it than normal to prevent those stalls. If you don't you'll get a quick lesson in stall recoveries that hopefully won't happen too close to the ground.

I think it's fun to get up in 15-20 mph wind with my Cub and see how long I can hold it in one spot, but I'm awfully busy on the sticks doing it. Beginners normally don't have the reflexes or the understanding of what the turbulence is doing to keep up with it, so your chances of crashing go up substantially on windy days. I was lucky/unlucky depending on how you look at it in that I learned at a field that's right by a lake where the wind is always blowing. I didn't have a lot of flying field days (still don't) in any given month, so if it was 20 mph wind on the day that I had set aside to go out I either could chance it or I could go home and wait 2 more weeks to fly again. I wasn't willing to wait, so while many other pilots were putting up their planes I was getting mine out and having a go at it even while still on the buddy box. I leaned to play the wind before I learned to do loops and rolls. But since you're learning on your own, develop the discipline to say "no" on those windy days and save your equipment. Those are the times when a foamy sport plane comes in really handy (the Parkzone T-28 is outstanding). They bounce well and are aerodynamically designed to cut through the wind better than a trainer does.