I am a fan of riders first and teams second, but I am a fan of both.
I became a fan of Unzué's teams through his teams being filled with riders I liked, and as a result of following those riders came to be familiar with the other riders on the team, and then now many of those riders that turned me into a fan of the team are gone, they have in places been replaced by other riders I like, and then they've suffered such extreme bad luck and had such a terrible year that I don't think anybody can begrudge them success; to whit - they lost Joaquím Rodríguez, a rider I really like, but signed Juan Mauricio Soler, who I also really like, and then Valverde got banned, but Tondó signed. I was a fan of Soler and Tondó long before they signed with Caisse/Movistar, and I remain a fan of J-Rod now he's at Katyusha.
The thing is, some teams simply have a certain aesthetic or a certain attitude that makes us like or dislike them. The DS/manager shapes the team around his aims (or which aims are available to him). So HTC, for example, become a loathsome, hated team for me, though they have some riders on the team who I quite like and won't begrudge their results; Farnese Vini/ISD-NERI, on the other hand, are a team whose entire philosophy for about three years was based entirely around animating races and making things more exciting, and therefore they are a team I like and enjoy. On the other hand, now that they're losing Visconti they will probably crystallize around Guardini and turn into a team that no longer sticks 4 of its 6 riders in attacks and turns a race that would otherwise have been a Greipel parade (Tour or Turkey 2010) into a half-watchable event, but instead contributes to the dull racing that I dislike.
Teams in cycling are still transient, evanescent things; they could fall apart and disappear in the blink of an eye. But how riders behave is shaped by their bosses, and who the team signs is shaped by that too. If you find a number of the riders you like riding for the same team you will like them; if you find a team who catch your attention doing things you like, you will pay attention to those riders and become fans of those riders.