There are a lot of friendships in Buffy although many of them aren't explored in any way. For example, Xander mourns his friendship with Riley but we didn't see much of that friendship before he left Sunnydale. The core Scoobies are friends all the way despite some unrequited (or not in the case of the fluke) feelings. The Trio are friends albeit traitorous, murderous ones. We see that briefly Tara and Anya meet each other without their partners. You could say that Giles and Buffy and the Mayor and Faith are friends although I think we should see their relationships as that of mentors or even fathers. Buffy and Faith were even friends for a little while but that went slightly awry. I've missed out more I'm sure but my favourite friendship in Buffy is that of Spike and Dawn in seasons 5 and 6. They got on really well with him looking out for her and her being able to talk him about anything.
ETA: After this comment from Page 48, I'm feeling a little bemused with myself that I didn't even think about Tara and Dawn's friendship which was pretty blooming cute. From thumb wars, to "knock yourself out", to Dawn sulking when Tara left Willow, to "good God, that's a lot of shake", to Dawn squealing with delight when Tara and Willow got back together, I'm afraid that Spike's going to have to shove off and be replaced with Tara.
ETA (again): I have just been reading about Buffy and Tara's friendship and, while it's not going to replace Tara and kittenface as my favourite, it was a friendship begun in mutual understanding and it had potential.