GA flight instructor with student reported a track heading deviation after turning to the wrong intersection resulting in a CFTT event. ATC advised the pilots of the terrain issue prompting the crew to take evasive action.

Narrative

Before attempting this specific flight; As an instructor; I had done this exact flight a few times both under VFR rules and IFR rules; My student and I were doing an IFR training flight. We had planned to go from PVU to DTA to U14 (Nephi) and then back to PVU. We were going to do 3 instrument approaches at each airport preformed under an IFR flight plan. After being cleared for the first approach (VOR 35) into Delta; the controller asked what we were planning on doing after either the missed approach/touch and go. We advised the controller that we were planning on doing the missed approach then we would continue to Nephi to do an approach; then head back to Provo for the final approach. The controller acknowledged it. We decided to do a touch and go at Delta to comply with the cross country logging. After consideration of the runway length; the winds and the high density altitude we decided to take it to the end of the runway and to takeoff in the opposite direction to preform the missed approach for safety precautions. Upon doing the missed approach; we advised ATC of the missed approach. We noticed it was a different controller; and the controller acknowledged the missed approach. We continued to climb in the hold until 9;600 ft. In previous flights; the controller either had us climb higher or had us climb after the hold. I assumed that we were good to leave the hold and continue toward Nephi for the RNAV 35 approach since that is would we had talked about. This is where I made the mistake as the instructor. Because I was unsure of the clearance or what was expected; I should of just asked the controller in the hold what he would of liked us to do instead of assuming that we were okay to continue to the next approach. After leaving the hold (because the DTA VOR is the IAF for the RNAV 35 approach) we were briefing the approach as we were going direct to the next fix; which was RAPPO. The controller came on and said; 'Where are you going?' We replied; 'We have began the approach segment for Nephi'. The controller was not happy and said that we weren't given a clearance to leave the hold. He advised us of the altitude restriction on the approach plate and advised us to climb to 10;800 ft and then he would be able to clear us for the approach. So we climbed to 10;800 ft and he cleared us for the approach.After having the approach already plugged in; we were unable to plug in the minimums. The small FMS knob was not functioning so my student used the FMS on the MFD (Multi-function display). Because there wasn't a way to only input in the minimums; my student just ended up re-inputting the approach on the MFD. Just a few miles before RAPPO; instead of putting the IAF as the Delta VOR and changing it in the FMS; my student put PILTE as the initial approach fix. This then took us from 070 to 050 direct to PILTE. Because we were so close to RAPPO; I had assumed that we had already crossed over RAPPO and now we were going direct to the next fix; PILTE. After picking up the weather for Nephi; I told my student to advise the controller that we had the weather for Nephi and that's when the controller asked us where we were going. My student then responded direct to 'PILTE'. The controller said that there was a low altitude alert that that we needed to climb because going direct to PILTE we needed a minimum altitude of 11;400 ft. So we climbed to 11;400 ft. We had already cleared the mountainous terrain at this point and by a lot. At that moment I knew there was some sort of confusion and something was not right. That's when I realized that by inputting the flight plan it changed. As we were getting close to PILTE the controller said; cross PILTE at 10;600 ft so we complied and began the descent. Upon doing the descent; we were given the phone number for Salt Lake Center. After so; we cancelled our IFR flight plan while doing the approach.Factors that may have contributed to the event/situation include: Time of day...it being very warm in temperature. As a CFI; this was my second cross country flight of the day. Perhaps I may have been slightly dehydrated or somewhere along those lines. To correct for this; I will always carry water with me even when the temperature is cooler. As a CFI; I assumed things when I should of taken initiative and not assume them. This was a learning experience for both the student and I. I will do my very best to not make the same mistake again. I take full responsibility as the instructor. Corrective actions that we could do would be to ask and verify if uncertain of anything. Also when replugging in the GPS to verify that it is doing exactly what it should be doing or just reference the minimums instead of trying to play with the GPS to input it. In conclusion; remain in the hold until cleared to leave. And if you are uncertain of the matter; just ask.