Light Transport aircraft Captain reported a track heading deviation during takeoff from SLC airport resulted in a CFTT event.

Narrative

I was cleared for the Fairfield Nine departure out of SLC; avionics systems checked that waypoints were in and FMS was showing the proper direction of flight before takeoff. I initially climbed the 160 heading; as prescribed in the departure. Salt lake departure then cleared me direct FFU. I put in direct to that waypoint as I normally always do; after FFU was entered direct the CDI needle showed I should turn left approximately 040 heading to the north. Unfortunately I proceeded to do what was shown instead of contacting ATC for confirmation. ATC shortly thereafter had me accelerate my climb for terrain. ATC received a terrain warning; hence the need for a climb; as there is high terrain near SLC to the east. However; my TAWS never went off warning me unfortunately. Shortly after I was put on a heading of approximately 230; I then went to the FMS and tried getting 'direct' to waypoints on the departure and at one point to the destination (ZZZ). The CDI never showed the proper direction of flight; however my proline 21 box did show the proper heading to the waypoints I wanted. I remember before taxiing 'initializing' the position of my aircraft and not receiving any warnings before takeoff. However; I may have initialized ZZZ as the position instead of SLC. I say this because as we kept getting closer to ZZZ (our destination) our distance on the MFD (Multi-Function Flight Display) was increasing when it should've been decreasing. Along with after trying to initialize a position in flight a yellow 'CHK POS' appeared. Also an hour into our flight the PFD (Primary Flight Display) 2 went into 'ATT/HDG (Heading) Aligning' which then turned off the autopilot; I hand flew for approximately 20 minutes and it came back online. I reported this to ATC while it was happening. There were numerous mistakes made; obviously I shouldn't have made the turn to the north; and clarify with ATC; I'm also wondering if the proper initialization ever happened. There were other issues that were happening with the avionics in flight to which I'm Unclear if it had to do with me possibly not giving the proper initialization or a separate issue all together.