Rory meets her idol Christianne Amanpour who I think gives her some encouragement (as she has been in the doldrums about her career). Anyhow she gets a job at an online magazine in Iowa and the show ends on the day she is leaving town with her and her mom having an early breakfast at the diner that Luke has opened especially for them.
As for their relationships:
Luke & Lorelei's relationship will be renewed (well, they were kissing in the preview!).
Logan is not in the finale but he is mentioned (don't know in what capacity or how much).