Quick Improvement and Racket Tips

High sophomore here- Learned tennis for a month 5-6 years ago, then quit because I was playing with kids half my age. Recently my friends who are lower seeds in the Varsity team challenged me that I couldn't make JV even if I tried. Being competitive, I agreed. I've played a little squash and badminton, where my strengths are less technique and more physical- namely running more than the opponent until I can hit a decent shot.

Tryouts come up in a month, and it seems like to make the team I'll need to be able to beat a couple of 9/10th graders, whom have already played tennis for a year. I'm a quick learner and am willing to take lessons/go practice daily in two weeks, when the local courts are cleared of snow and open to play. I've learned most the rules, a little forehand and backhand technique from these friends, but haven't been able to hit any balls yet. I'm also looking to get a decent racket to last the season or over the summer, when I can play with friends (hopefully I'll have gotten better by then), I've saved up around $50 and probably will get up to $100 by then, so budget constraint: up to $100.

Please recommend beginner rackets and tips to improve quickly to a semi-competitive level! Hopefully that'll get me through tryouts. (I think learning how to serve will be a big part of the evaluation)

Thanks!