How to view the photo settings on my Nikon D5200?

Like, after I take a photo, I want to see the shutter speed, aperture, ISO settings, straight from the camera.

usually on my old camera, the D3100, I just press the up button while viewing photos. But on the D5200 it doesn't seem to work.

All of your answers are in the manual.

Press INFO?

Check the manual.

Press the INFO button:

The D5200 is similar to the D3100 in that all you have to do is press the up button when viewing a photo. What may be confusing you is that the D5200 has seven different choices. So just pressing it up once may not be enough. Look at one of your photos on the camera. Press the up button seven times to see all the choices you have.

Your camera's manual is a wonderful resource for answering such questions and is much faster than asking online.