First off as a Corp we're very friendly to noobs. We are at war currently which depending on your outlook isn't the best environment for a new player to start the game, although it does highlight Eve's status as a sandbox game and one where if you build a sandcastle someone bigger than you is going to try and kick it down for no apparent reason, so if kicking down other peoples sandcastles appeals to you read on.
Secondly how long it take is partly down to what you want to do as a group:
PvP? A few days to start contributing either as a tackler where you hold someone in place while others damage them or using electronic warfare to stop enemies damage your friends within a couple of days. Don't expect to setting the universe alight straight away though you'll lose plenty of ships, but it's fun as long as you: "Fly what you can afford to lose" which should probably be emblazoned on Eve's login page.
Mining? Well there are mining frigates/cruisers to use or industrials to help transport the ore that would take under a week to train for, but ultimately you'll want to get yourself a mining barge which would take a couple of weeks.
Missions? There are plenty of routes to take here you could help a corp mate in the large Level 4 missions through salvaging or join a group of people for Level 2/3/4s to get the done faster all of which would only take you a few weeks.
Eve seems intimidating from the outset, but complete the tutorials as well as the Epic Arc they send you off on and you'll find you have a decent chunk of isk in your back pocket and can start trying things out to see what you enjoy.
Free hats are available and tailored to your specifications. I myself have a fine pink, fur lined cowboy hat that strikes fear into the hearts of my enemies