I used to develop games and if you are doing this, then you will want to be very very patient.
Having said that, I think you could develop a simple game pretty quickly. I wouldn't try and make a 3D game, but I do think a 2D game is doable within a short time frame.
You will need to have a "mathematical" mind for this. When I say "mathematical" I don't mean that you will need to know pure mathematics, but you will have to translate things into math and data structures and then take that and implement it using code.
A simple chat program is a good way to learn networking protocols. I don't know what Python supports, but I would be surprised if someone hasn't written a network API for it. (I don't program in Python by the way so I don't know).
Making a game can be really painful, but when it works its a good feeling. Just so you get an idea, it will take you many many years to make a good 3D game engine which is why a lot of people just get an existing one (whether its free and open source or closed source and commercial).
I'd think about either a really simple 2D game (like Snake where you are a snake and gobble things up to become longer and longer) or a chat program where you decide how many features you have based on how much time you have remaining after doing the absolute basic version.