"I purchased a computer from this small local computer shop. I paid them to build me a computer for general purposes (surfing the web, watching videos, etc). I could have done it myself but prefered paying to have it done painlessly. "
Okay
"About a week in I start noticing some issues with videos online lagging a lot and such. So I looked at the specs of the computer they made for me and noticed it only had 4G of ram. This seemed a bit ridiculous for a modern computer. "
Hold on. You didn't make any requests about the specs of the computer when you decided to purchase it? Surely they gave you a copy of the specs of the machine or at least made you aware This doesn't add up.
"So I receive it, open up the computer and remove the old ram. And then just like that, a spark appeared and nearly killed me. "
Perhaps you should've either turned off the computer first or unplugged it first. That would be user error in either case.
"I know I installed it myself, but the PC was inadequate to begin with. "
1. Your fault for not knowing what you were doing
2. Your fault for not knowing what you were buying
"And it should have never blew up like that?"
If you had turned it off first and/or unplugged it, that wouldn't have happened. They shop wasn't negligent, the user was.
" I'm beyond annoyed at them at this point. "
With yourself, I would hope. You bought a functioning computer, that's all shop is responsible for. Once you took matters into your own inept hands, you assumed liability for whatever happened next.
" What can I do?"
Replace the motherboard and/or RAM and/or CPU (if needed) at your cost.
"Do I have grounds to sue them?"
Sue them for what? They sold you a working computer. You apparently did not make it known that you wanted 8GB of RAM when they built you the computer. Then you started monkeying around with it (with as it turns out, an inadequate level of expertise), you break it, and you think that it's somehow the shops fault? That's staggering. Go ahead try to sue them, and see what happens. If you're lucky the shop will only counter-sue for their legal costs.