I have a epiphone les paul jr, the same that the game comes with. For a while my high e string wasn't staying in tune. After switching out a few strings I found out that it wasn't the string and that the tuning peg was loose. How do I go about tightening the peg so that it stays in tune? Thanks

Did you wind the string on properly?  High e strings like a good few turns around the peg.

 

How long are you giving it before deciding it won't stay in tune?  Strings can take a day or two to settle in.

 

 

Cheap tuning pegs are cheap tuning pegs, maybe just buy some better ones...  or keep tuning it, it's all good practice :D

Some models like my Gibson SG and Epiphone have a screw at the end of the knob. Try tightening it a little, that will stiffen the mechanism a little and it can help. Worked for me