I'm more of a fan of Fate/Zero mainly because of Kiritsigu and it was just an overall darker theme. If you haven't watched it yet I highly suggest it before continuing with Fate/Stay Night as it is a prequel. However most people say it doesn't really matter which order you watch it in. Whether you want to watch it as a prelude or prequel.

I hated the original Fate/Stay Night and I mean absolutely dreaded it. However Ufotable, the creators of Fate/Zero anime, saved it and made me like it. However it's still not on par with Fate/Zero which was fantastic.

As for your questions there are quite a few character. I'm not sure where you are at right now so I'll try and avoid giving you spoilers. There are 7 heroic servants and their masters. You've met Shirou and Rin right now which are the two main protagonists. Then there is Kotomine who you've met briefly in the church (presuming you've watched at least 2 episodes ). Again I'm not sure if you've watched Fate/Zero yet so I won't go into his character yet. All I'll tell you is that the character are very interesting. I find Caster to the most interesting servant and maybe even character of the show. Although Beserker and Illya are also my favourite. Again that's more because I've watched Fate/Zero.

I don't really read the manga so I can't really tell you my favourite chapter. Fate/Stay Night hasn't finished either so I can't really tell you that anything epic has happened yet. However the final episode from Fate/Zero was fantastic in my opinion.