Hey there, one of the best ways to level up is playing some ranked round tournaments, like the weekly 9 hole unlimited. For 9 holes, It usually takes anywhere from 15 minutes for faster players to 25 minutes for someone like myself.
You do not have to belong to a country club, but it could help out with your game. Most people in a club will help you out with tips or advice while playing.
I wouldn't be in any hurry for new better golf clubs untill you are more familiar with how all the clubs work and perform in different situations.
Hope this helps, and good luck.