As my car had problems with the transmission, I had the clutch plate replaced on it. After the repair was done, I started it up the following day and drove about 25 miles when the issues started happening again. The transmission started to slip out of fifth gear. Then, it wouldn't go into fifth at all. I drove about ten more miles when I noticed a grinding sound coming from the transmission. How serious is this? Is my transmission going out?
This is possibly a transmission shifter adjustment issue (5th gear, non-engagement). The noise issue may be related, but it is potentially serious regardless. You don’t mention what kind of "transmission problems" caused you to have the clutch repaired, but that problem may be related as well. Regardless, you should have the slipping issue inspected by a professional, like one from YourMechanic, before driving it further.