If you want into PL AFAIK you used to need to join Snigwaffe.
They are/were the official training corp. I knew a guy in there and they sounded pretty organised and solid. Not sure how it works now? Didn't Waffles leave PL?
GF it used to be the case that you had to be a real member of the SA forums.
They have ancillaries like "Merch Industrial" from the Penny Arcade forums that are easier to get into but GF proper was a forum member only thing. They encourage new members to scam pubbies for referalls so don't give anyone ISK.
Dunno if any of that is current. Pop along to their public channel and ask / hang out and read is probably the way forward. By being in a prospective corps public channel you show interest and get a feel for the culture and whatnot. It's good form whoever you want to join.
You're welcome to apply to SOTF and I will put in a word for you. Can't guarantee you'll get in because I am not the HR guy (that's Galus Mendak) or a director but I will certainly attest that you are not a tool.
We're a PVP corporation. We started out in FW but we've branched out a bit now.
Presently we're holidaying in Messoya to molest the locals. Sensing a fine opportunity for murder, mayhem and extortion have popped a roof rack on the jump freighter, stocked up on sports mixture and gone on a road trip.
Our PVE activities are not super organised. We're in FW so you'd have access to FW missions. We also do other stuff like deadspace plexes and a bit of PI / industry but I have to emphasise that we are a PVP corporation. We're not a mission running corp that does a spot of PVP or a sov holding corp that bears all the time and reluctantly turns up for CTAs. We live to ruin pixels. Most of them will be other peoples, some of them will be ours. Our CEO is an acoholic yorkshireman. Our best FCs are vodka-fuelled.
In terms of help, we have an excellent logistics chain with carriers, JFs and a web based corp shop (like amazon for guns and ships). Depending on what you have and where we can probably help you move.
http://www.shadowsofthefederation.com is us.
edit
The Mittani stepped down because some miner he suicide ganked threatened to top himself.
The Mittani then encouraged everyone to troll the guy at a Q&A in fanfest and gave out his character name. This was arguably cyber-bullying. Either way, CCP don't want to be appearing to encourage this so he's stepped down.