Alright, so I got this cheap little (less than 2 feet long) guitar. It was $10... and yes it's a piece of junk.
BUT I want to fix it up so that I can actually play it. The original strings were absolutely terrible so I got some Extra Light D'Addario strings. Now as I was tuning these strings, I noticed they kept going flat, and eventually I head some cracking sounds. The bridge seemed to be coming off. Also the nut was being silly too.

Is there some good wood glue or other resin I could apply to the bridge and nut of this thing so that I can tune the strings on it?
I'm not sure Krazy Glue will do the job...

I DO REALIZE THIS THING IS A PIECE OF JUNK. But its cool! and its cheap!
Mini guitar $10, Strings $5, glue/adhesive $??
If it worked it'd be awesome.

0827002357 by rickyx817, on Flickr
i left one string on for novelty purposes

0827002358a by rickyx817, on Flickr
little plastic nut... elmer's glue maybe?

0827002358 by rickyx817, on Flickr
the bridge seems very weak
You mean the saddle is coming loose? The bridge is the black wood block. I would use super glue on the nut. Idk about the saddle. Maybe you should be using nylon strings.
That is a nylon-string bridge; it may well not be able to stand the stress of steel strings.

It appears that the saddle is very poorly fitted and possibly the slot containing it has cracked as well.
I would try removing the saddle and making sure the slot and surrounding wood is sound.
If it is cracked, you can introduce super glue into the crack and clamp it. Then you'd have to re-fit the saddle.
Odd looking instrument... A child's guitar? A six-string uke?

The nut is normally held in place by string tension, little more than a drop of glue to hold it in position is plenty.
Yeah, its wood.

I didn't think about that. It came with steel strings, but they absolutely terrible.
Do nylon strings have significantly less tension?

Yeah, its ukelele sized I think, but its some made-in-china guitar probably kids guitar.