Not sure what the problem is, but after a few hours, you should definitely be able to save!
If I understand your first post correctly, esc doesn't do anything (not even take you to the main menu?), which is definitely odd (I just tried a new game and esc takes me to the main menu as early as the introduction).
You should post your problem in the game support forum
and maybe add a little bit of info like what computer you're using (PC or Mac) and which version you're playing (Telltale or Steam).