Well, since I've never played Tsukihime, I can only talk about Fate characters.
My favourite character is, obviously, Sakura. I just find her mental strength to be amazing, as well as the way that she remains kind and caring despite everything she's been through, and the way that she is able to bounce back so easily after being freed from Zouken's control. Plus, I find her extremely sympathetic, particularly since she is basically ignored in two of the three routes, and has to go through a hell of a lot to get her happy ending in the one route where she does get it (and a lot of people in the fandom still seem to begrudge her that one happy ending).
As for the rest, I like Rider a lot, and also Shirou, particularly the HF version (and excepting MoS Shirou, who is an asshole). I also like Kariya, and sometimes Rin (particularly post-HF Rin, who is less of an idiotic magus in awe of her idiot father).
Rider I like because of her loyalty and protectiveness towards Sakura. Not the stupid, blind sort of loyalty and obedience to her master out of duty that Lancer has, but rather a genuine loyalty borne of real compassion and love for Sakura. She won't always obey Sakura, but she'll do what she thinks is best for her, and will protect her no matter what. And, whilst she does some bad things sometimes, she's basically a good person at heart, although she can be pretty nasty towards people she doesn't like (which is something I can relate to, because if someone treats me like shit and makes no effort to atone for that I can get vindictive). Plus, she's really hot and her sexual preferences align with mine, which is really handy when it comes to looking at, reading and writing porn.
As for the rest, Kariya and Shirou I like because of their determination to protect people, and Sakura in particular. Both of them were willing to sacrifice everything to protect her, even if it meant their deaths (Shirou might only actually act to protect her in HF, but I suspect that Fate or UBW Shirou would be just as willing to die to save her as HF Shirou was, even if the romantic relationship wasn't there). Indeed, in the case of Kariya and HF Shirou they outright expected to die, and yet they still continued.
In Shirou's case, even outside of HF I do still like him. I do find his ideal (in the pure form) admirable, even thought I detest the corrupted "kill one to save many" approach that MoS Shirou takes. I know it's not achieveable in practice, but it's admirable for him to try. I am someone who likes idealists and people who do what is right rather than what is convinient. And, whilst Sakura gets completely screwed over outside of HF, it's hardly Shirou's fault, he knows nothing of what is happening.
Finally, there's Rin. My opinion of her does vary somewhat. At times, she's really likeable, but at other times (particularly around the MoS part of HF) she's an outright bitch. Still, as irritatingly-lawful as she can be at times, her heart is fundamentally in the right place, and when she's not going out of her way to hide that she is a likeable person. My opinion of her depends somewhat on the route (like I said, post-HF Rin is much more likeable due to ditching her father's "true magus" crap in favour of her sister).
Also, Rin is odd in that she's the one character who my opinion of has actually changed quite a lot over time (Saber too, to some extent, in the opposite direction, but I think that's mainly because of how popular she is and how much Type Moon milks her). When I first completed the game I really didn't like her at all, because she was just a complete and utter bitch to Sakura for most of HF. But, as I talked to people and thought about it more, she came across as a much nicer person than at first glance, even with regards to her sister. Plus, she's a victim of her upbringing just as much as Sakura is, in a way, so it's hard to completely blame her for how she came out.
