Savor the first time then go back for trophies. On chapter 9 on my third Uncharted playthrough now and going for the speedrun trophy. Still gotta get to Doom. Unless I'm going for trophies I normally play on easy and enjoy the story especially the first time.

Savour it, every last minute of play I can get, if it's a great game I'm loving I'll be the last to finish it, I'm often the last in my guild to the max level in a new expansion or currently at 70 hours in Digimon Cybersleuth just grinding up digimon into their Mega forms early in the game... far earlier than I should have been doing it but just enjoying filling out the Pok.... Digi...Dex!

Depends on the game. If I love it, like a Zelda, or Uncharted like you mentioned, I savor the moment and start doing everything you can do. That doesn't even limit itself to a first playthrough though, I'll happily spend 60 hours on KotOR again for the fifth time. Otherwise, if I start to lose interest in a game slowly, like when I feel a game starts to drag, or when I don't really like it but still want to see the ending, I speed things up and start ignoring anything on the side. If I'm not interested anymore at all for whatever reason, I just quit.

Unless I have a time restriction, I play at my own pace. Partly why I despise moments in games that require rushing through, unless that's the entire point of the game. I'll probably never platinum an Uncharted game, purely because I hate speedrunning. That said, I'm not one to drag my feet to artificially extend playtime, either.

I only rush through backlog games that i've lost interest in.

I never rush through story based games, if i'm enjoying them. I take my good ole time, and search as much of the map, that i possibly can, for the hidden stuff.

I just try to get whatever achievements I can get and play with friends games for fun.

I only rush games I have already finished before.

It's awesome. I did my speed run in cell shaded render. I originally was going to do it in 8-bit mode, but it's almost unplayable. You can't see shit lol. 

If it's a game I've been waiting for a long time, then I would definitely not rush through it, unless there was some time constraint (like I have to return the game or something)