I'm definitely a novice when it comes to cutting a corner such as like I'm trying to do here. I'm want to put some quarter round trim at the top of the existing trim near the ceiling. However, I really need some help in figuring out the angles I need to cut the ends to meet up properly at the corners. Please see the pics below. Thank you!!

 

If the corners are reasonably close to 90° you cut the quarter round 45°. If this is new construction not must effort is made to get the corners square so coping may be a better way.



When you do the cope cut the angle slightly deeper toward the back to compensate for any out of square of the piece your overlapping.

Splices are done with 45° cuts.

 

It's not new construction. How exactly would I "cope" the second piece? How difficult is it? Have any advice?

 

Coping will give better results then trying to just miter the joints.

Google "How to cope an inside corner" Or "How to cope"

This might also help: How to Cope Joints: The Family Handyman

 

Are you just putting 2 pieces together along the top... or three?

If you are doing this on a miter saw, you set the saw at 45° right, for one... with the quarter round on the right of the blade and 45° left for the other one... with the quarter round to the left of the blade.

I am guessing that you are just having problems positioning the trim on the saw. You need to imagine that the base of the miter saw is the ceiling, and the fence of the saw is the wall, and that the pieces are upside down as you cut them.

You also need to have the trim sitting the correct way on the miter saw. With that back bevel it might make it a little more difficult for you to figure out than if it were totally solid quarter round. Maybe it would help you if you put some masking tape on the fence... 3/4" above the table. And also a strip on the table... 3/4" away from the fence. This will give you a visual reference for where the moulding goes on the saw. It needs to sit at a 45° angle as you cut it, just as it will be sitting in the upper corner of the ceiling. (but upside down)

If you don't want to miter them, you can always try coping them. I'm not sure that learning to cope is a good idea right now since it will probably confuse you even more, but you can certainly try it.