So I just got Diablo II for PC, (Coming from Fate) and I'm not sure what character to use or where I should dump points. And I'm not entirely clear on Rune, rock... Things.
Any help would be appreciated!

I'd say this, a Character Build on the internet would surely help you, look up some different ones and follow their directions, you'll eventually catch on what they'll make you able to do!
f.x you chose a sorceress and want to be focusing on using fire spells, find one that does that, or do one that does ice or lightning etc.
Lightning is by far the most used sorceress type btw
ignore all the info on the builds about which type of weapons and armors etc. you should use, cause these are legendary items you won't get anytime soon
Just follow the build, learn to use it, and it'll work out!

Runes:
Runes are used into socketed weapons, and IF they are put into the weapon in a certain order, they give some special abilities! can't remember the name for all runes,
but an example would be "Sur-Lo-Sol". That would mean putting the Sur rune in first, the the Lo, and then the Sol.
Look up "Diablo 2 Runewords" on the internet and find out how to put runes together into the weapons.
REMEMBER: The "Ladder Only" runewords won't work in singleplayer gaming, so stay clear of trying to use them!

Thanks! I was going with a melee guy though, wasn't sure to go with barbarian or paladin.

Then look up some builds that will go with what you feel like!
With a Paladin, a Hammerdin build is very strong and quite popular, he uses a spell that makes a holy hammer that circulates outwards from your character and hurts all in its way, you just spam that like hell!
Else look up a Zealot build, he uses a speedboost that makes him hit people amazingly fast while using a skill that gives bonus to different elemental damages, that makes it a great attack against any enemy as a creature cant be immune to all those elementals at the same time