Hello Stevanator!
Most folks start in a Hyperion, good place to get into incursions.
Getting in: At the top of this browser window, next to Forums, is 'Waitlist'. Clicking that takes you to the waitlist, which is what you use to get into fleets, when there is one running (which is the majority of the time). Next to waitlist is 'Fits'. Check out Hyperion in fits, and get a Hyperion fitted that way to the current HQ system, which is Todrir (just moved here, so good for a few days, up to a week). Then 'X Up' on the waitlist, and when there is room in the fleet, the FC will get you an invite and a newbro speech.
There is not a singular Corp - rather, join the "Warp To Me Incursions" channel - mentioned in the Rookie Guide. When fleets are getting started it will generally be announced there. Or, watch the waitlist site. Also a good place for incursion questions!