I've been playing bass for less than a year now and decided to borrow a guitar
from my grandpa and began playing it about a week ago.

It's an acoustic, and whenever I try to make chords it's as though my strings are trying
to rip through my skin and when I lift up I see crazy indentions (sp?) in my fingers.

Will this eventually go away?
I already have decently rough fingertips from playing bass, so why does it still
kill my fingers on guitar?

Also, certain chords tend to make my fingers hurt from the placement, is it because
I'm placing my fingers badly or is it because I just started playing?
Strings hurt like a son of a bitch when I started. It goes away in a month or two. Keep playing, your technique will usually improve by itself.
--"Hey, don't worry; don't be afraid, ever, because this is just a ride."--
It will hurt more on guitar, because the strings are thinner.
It will go away in up to a month I think.

It may be because they are badly placed, you might want to ask UG for the fingerings of ceratin chords.

It IS more likey because you've just started playing.
Bass forms different callouses than guitar, because of the different size of the string (I think), so once your fingers start to get used to the guitar strings, the pain will go away.

What chords hurt to play? The only chord that I've had discomfort playing is barre chords, due to the pressure on the wrist, but that eventually went away. Anyway, if you're playing something incorrectly, after playing a bit you tend to automatically correct what you're doing.
Yeah, it's perseverence really - probably a good idea to not play through pain though, so stop when it hurts. Take it slowly, and it eventually stops.

When you get to barre chords, it begins again...
The finger problem is definately b/c of the thinner strings. If you want, you can move up a size or two if you think itll help. You will eventually get callouses, and your fingers wont hurt so much. The only chords that hurt to play for me are crazy chords, like add chords and slash chords (G/F# for example). But those are rare, especially if your a beginner.
Yeah, I'm like you picked up a guitar after learning bass, It will eventually go even if it does hurt so much when you first start just carry on playing it will go away dude.