General aviation pilot reported receiving a low altitude alert from ATC and experiencing a CFTT event during departure climb. Pilot complied with ATC instructions and avoided terrain.

Narrative

I was departing SQL on an instrument flight plan. I made a turn on course before being directed by ATC. In my self-examination of the issue complicated departure instructions including unclear delineation of VFR/IFR; and unfamiliarity with the area; contributed to a pilot error in single pilot IFR. I am an experienced IFR pilot; still getting used to SQL and the surrounding airspace. I have flown into and out of San Carlos Airport multiple times; including today; within the last 90 days. This was my first IFR departure to the south-east.Since this was a navigation error in controlled airspace under IFR my navigation suite should be described. I am using a Garmin GTN 750 coupled to a Garmin 1080 G500 TXI (Touchscreen Flight Display Engine Information System) PFD (Primary Flight display); I have a Garmin SL30 NAV/COM coupled to a VOR/Glideslope as a secondary nav. San Carlos has implemented departure procedures because of noise abatement concerns and the proximity to the SFO BRAVO and the hills to the west. The departures are pictorially presented on a sign next to the gate to the terminal; it is a complicated visual listing: the Hillsdale Departure; the Oracle Departure; the Coyote Hills Departure; and the Woodside Departure. Each has several visual waypoints and familiarity with local landmarks is required. My experience when VFR is that when I am handed off to ATC for flight following and identify e.g. 'Aircraft X at 1;000' on the Hillsdale Departure'; ATC has no idea these procedures exist. This is important because if I had been more familiar with these departure procedures; or if they had been in the form of waypoints or a SID the GPS would recognize; the memorization required to transition from the VFR departure procedure to the instrument portion of the flight plan would have been smoother reducing workload. Today I filed a flight to ZZZ to the southeast. I received my clearance in the run-up area before departure.I copied the following clearance (typical CRAFT [clearance] format copied here in the sequence given). Note it took two tries to capture the detail and get the readback correct:Cleared: to ZZZRoute: Fly runway heading until you pass the diamond-shaped waterway; make a RIGHT turn to 120 degrees; keep downwind within 2nm of the airport; (Altitude:) maintain VFR at or below 1;100' until passing the Oakland (OAK) 165 radial (which is roughly abeam the approach end of Runway 32) then Vectors Woodside (OSI) [VOR] remain at or below 2;100' expect 5;000' five minutes after departure.Frequency: 135.65Transponder: XXXXI programmed the GTN 750 with the ATC routing (SQL OSI SJC ZZZ) and programmed the secondary radio VOR to the OAK 165 radial.On departure; I flew the Runway 32 heading and climbed to about 300' MSL; I identified the diamond-shaped waterway and began my right turn to 120 degrees; a 180-degree turn. I finished the turn within 1 nm of the airfield and arrived at OAK 165 about 1nm east of the approach end of Runway 32 at 1;000'. Up to this point; I believe I am still flying VFR in a departure procedure; despite being on an IFR flight plan and having received an IFR clearance that included 'maintain VFR'. At this moment I was handed off to Norcal Departure and checked in; 'Aircraft X; 1;100'. As I glanced at the GTN I saw the magenta GPS line to OSI passing off to the west. Looking back I think Norcal responded; 'Aircraft X climb and maintain 3;000.' I had been expecting 2;100 based on the clearance. I corrected my altitude clearance bug to 3;000.What my mind heard was 'Aircraft X climb and maintain 3;000; direct Woodside' At this point I had mentally discarded the 120 heading as part of the VFR instruction and made a climbing turn direct to OSI; approximately a heading of 220. (This was my mistake. I did not get a new vector; I should have maintained heading 120. Getting vectors for a climb is very standard stuff; as are VFR climbs on course to the MEA - why did I screw itup?)A few moments later ATC said 'Aircraft X climb and maintain 5;000' I was not yet to 3;000 and seeing the mountains ahead; I updated the altitude bug and responded 'Expedite climb to 5;000; Aircraft X' and configured for maximum climb; still direct OSI. The mountains in front of me were tall but I was making good progress overcoming them. (The tallest point near the OSI VOR is 2417' and I was at 2;000' with over five miles to go.)A few moments later ATC called 'Aircraft X Altitude alert; MEA is 4;000' you are not on the assigned heading; verify that you can provide terrain separation.' I immediately realized my error. I responded; 'I apologize for my error; I can provide terrain separation. Aircraft X'. I waited for further instruction: Did he want me to return to 120 or maintain 220? I had crossed 2;000'; then 2;400' and there was no terrain risk. A few moments later ATC came on and stated 'Aircraft X; Emergency terrain separation alert; turn to heading 100; south lake' I responded '100 degrees; Aircraft X' As I turned to 100 I crossed 3;000 still five miles from OSI. The rest of the flight was uneventful. Thinking back on this; my mistake was in trying to memorize; visualize; and proceed through an unfamiliar departure. My brain discarded the heading of 120 as being a part of the already completed VFR departure procedure. I failed to sift it out and carry it forward as the assigned heading for the IFR portion of the flight. All of the other descriptors about diamond-shaped waterways; turn downwind with an offset from the airport; OAK 165 radials intersecting the approach end of Runway 32 were getting checked off and cleared from the 'memory buffer' as I passed them - the 120 heading from way back by the waterway was also discarded. If even one or two of these could have been covered as a waypoint; or better yet in a SID; then the number of items to mentally manage would have been less and the important detail would not get overlooked. The mental prompt of seeing the SQL-to-OSI magenta GPS line going by would not have prompted me to turn on the course. The conflicting 'maintain VFR' as part of the IFR clearance created the uncertain point at which I was unsure if I was in a VFR climb on course to the MEA or following vectors for a climb.Had I done this successfully; I should have checked in with ATC 'Norcal departure; Aircraft X off San Carlos; IFR; 1;100 on a 120 heading.' Clear and unambiguous. If only it had been presented this way.