The Tully clan doesn't play a major role in this book, but since Catelyn Stark was once Catelyn Tully, we do hear about and get to briefly meet her family.
For instance, Catelyn's father Hoster Tully is a dying old man, but from the stories we hear about his fights with his brother Brynden Tully, we can see where Catelyn gets her stubbornness.
Catelyn's young brother Edmure Tully provides a good contrast with Robb Stark: Robb is younger but better at war. By contrast, Edmure is too impetuous. Think about the time when Edmure wants to attack the Lannister raiding party led by Clegane. As Eddard thinks of it, that's just the mistake that Tywin wants Edmure to make (44 Eddard 11). So, Edmure ends up being captured by the Lannisters, while Robb ends up capturing Lannisters. Guess we know which tactic works better.
Perhaps our favorite Tully, and the one we see the most of, is Brynden Tully, Catelyn's uncle. Brynden is something of a black sheep (or as they say, black goat), which is why his nickname is "the Blackfish" (the color black + the Tully family symbol of a leaping fish). Brynden is an excellent knight and planner, always one step ahead of Catelyn, as she quickly realizes (60 Catelyn 9.17). Brynden is a lot of help to the Stark/Tully army.
But he's also interesting in this book because of his fight with his brother Hoster. See, Hoster arranged a marriage for Brynden, but Brynden refused. So, whose side are we on there? It makes a lot of sense to support the guy who refuses to go along with an arranged marriage; but at the same time, we have to recognize that marriages are not just about love here, but also about politics and alliances. Brynden and Hoster's fight tells us about this world but also presents us with a classic Martin dilemma, because we're not sure who's right.
Since this isn't a book about the Tullys, we don't meet a lot of their direct servants. We do briefly see Utherydes Wayn, the steward of the Tully castle, Riverrun, but not enough to get to know him.
But we do meet several knights and guards who are sworn to noble houses that are sworn to the Tullys. That is, when Catelyn sees Tyrion at the Old Crossroads Inn, she goes around the room asking the soldiers there if their lords are still loyal to House Tully. That's how Tyrion ends up being guarded by House Bracken soldiers like Kurleket, Lharys, and Mohor; and Ser Willis Wode, a knight in service to House Whent. Not a lot of these guys survive the trip to the Aerie, though.
Catelyn's group also includes some mercenaries, like Bronn (who switches sides) and Chiggen (who dies along the way). And don't forget about Marillion, a wandering singer who goes up to the Aerie (how did he ever survive the trip?) and decides to stay with Lysa Arryn. He's annoying, and never does anything right. This helps remind us that we like Tyrion (because the two don't get along).
House Tully is the primary house of the Riverlands, and they have several houses sworn to serve them. According to the Appendix, those houses are Darry, Frey, Mallister, Bracken, Blackwood, Whent, Ryger, Piper, and Vance. Now you may remember some of those names, because many of those houses decide to swear their allegiance to Robb Stark as the King in the north. So check out the Stark Bannermen for more on them.
(House Frey has its own page, though: even though it is sworn to House Tully, they can be a little – just a little – untrustworthy.)