I'll second @pang4's advice on the runner side:
Playing Adam is great for learning fundamentals of what you want to do as a runner (run every turn, trash everything, access cards from HQ). You will also learn how painful it is to go to 0c, but more importantly you will learn to bounce back from it quickly.
Playing as cams Hayley is fun, and reasonably easy to do on Jinteki (it's easier to keep track of all your abilities, credits, and 0 time spent shuffling). Since it's come up several times today, I'll link my cams deck again.
Siphon spam is a really strong play style. I don't have a good list for it, but basically what you want to start off with in Criminal is 3x Account Siphon and 3x Same Old Thing. Alternatively you could play Anarch Siphon Spam with 2x Account Siphon, 3x Deja Vu, and 3x Same Old Thing. Probably in MaxX, since the recursion works really well with her game play anyways.
I'd also like to suggest Apocalypse Kate. To learn the value of board state. The only thing you need to know about apocalypse Kate is you should always be playing Apocalypse if you think you can actually land it.
On the corp side, I think playing dedicated rush decks can be instructive, but you will probably lose a lot of games. I've tried NEXT Design a number of times, and I just don't think it's a particularly good ID. Titan can be a lot of fun, but Weyland also has a hard time winning. I think most of what you learn from playing rush decks is you really need something more late in the game to score the last agenda. I'd try to find a modern Foodcoats deck to try out, that fits into more of the never-advance game plan/tax the runner out, but it has been made much weaker by Rumor Mill.
PE is a lot of fun. Learning how to game what the runner is thinking is fun, but it's really best if you can play it in person to hear from the runner what they think is going on. I used this deck about 6-9 months ago and enjoyed it a lot, although at 11 Agendas, I don't think it's what @pang4 had in mind.