Chess calculating ability

Hi folks,

I'm quite new (a month or 2) to chess.com and am still exploring the magnificent world of chess. 

I was wondering whether you could give some advice as to how to improve my play as fast as possible. I hear from everyone that playing games is the best way to move forward. I'm playing a lot of games but don't seem to feel myself getting stronger. Is it necessary to start analyzing my games to improve? And how do you go about this kind of thing?

Also, how important is chess memory for gameplay? Would it make sense to try to 'read' games without a chessboard, just imagining it in my head?

If you have any other tips, please let me know..

Thanks,

Dudejan