You’re right that most of it is down to specific effects - most notably IMO “phasing” or “flanging” as it’s known, where two copies of the same signal are put gradually out of phase by very small amounts. Examples are Itchycoo Park (Small Faces), Nathan Jones (Supremes), and something else I can’t remember… (hey it was the 60s, man…)
The effect the Beatles (Lennon especially) liked was backwards tapes, and various other audio mangling techniques - then at the forefront of the technology (double tracking, various speed manipulations).
The idea is disorientation, so in terms of harmony it’s about surprising chord changes, and sometimes rhythmic effects.
Hendrix’s Burning of the Midnight Lamp is a classic. Firstly there’s the exaggerated burbling wah-wah in the intro, with a pseudo-exotic melodic embellishment. The chords seem to be in C, but end up as a V-I-IV-V in F major.
Then the chords take you on a trip: first a classic doo-wop I-vi (F-Dm) - but then it plunges you down another minor 3rd (chromatic mediant) to B major, resolving to E. Just when you’re starting to think it’s changed key to E, it goes to C major (chromatic mediant again), then G and D (phew back in normal-land), then F (hey chromatic mediant no.3! or just a rock bVII), before ending in G. The G then leads (logically enough) back to the C of the intro.
This suggests that chromatic mediants have a big part to play in any psychedelic chord sequence, but Hendrix handled them quite cleverly, always in balance. Typically in rock music chromatic mediants can just sound like borrowed chords. E.g., Ab in C major is bVI borrowed from C minor. E major, meanwhile, would usually be V of Am, or resolving to F in a deceptive cadence.
The psychedelic trick is to not allow the listener to settle on a key centre at all - as soon as it begins to seem clear, upset it. That’s what Lennon does in Strawberry Fields Forever, perhaps the archetypal example of psychedelia. The intro begins in E, with a traditional descending line cliche but then settles comfortably into A major at the beginning of the verse. But the first chord change plunges you into Em - “let me take you down” indeed. Then it descends further into the rabbit hole via a F#7b9 chord (are we heading for Bm maybe?) which resolves to D - twice - before settling on A again, but not before a 6/8 bar has been inserted.
Naturally this is all supported by George Martin’s masterful score - managing to make the Em and F#7b9 way more dramatic than any jazz tune would make them. (“Hey man, it’s just a secondary ii-V of Bm, what’s the fuss?“)
A chance occurrence adds even more psychedelic effect, due to Lennon’s demand to have two takes in different keys and tempos spliced together - which was managed by speeding one up and slowing the other down, resulting in a spooky change of tonal quality to voice and drums in particular (about 1:00 in). Even spookier is the outro to the song, where it fades back in like a dream flashback, with strange time shifts and overdubs.
George Harrison was also pretty good at various “psychedelic” harmonic effects: check out the 7b9 chord in I Want To Tell You, the dim7 and lydian effects in Blue Jay Way, the throwaway nonsense chord sequence in the bridge of Only a Northern Song.
As always, it’s a careful balance between those “weird” chord changes (help, what key are we in!) and the studio processing effects.
Of course, all these kind of effects (and others employed by Pink Floyd, Zappa, etc) had much more impact back then. Now they’re all much more “sweet and nostalgic”, gently amusing - psychedelia as just one more vintage genre to be played with.
More inspirations:
https://www.youtube.com/watch?v=DuABc9ZNtrA (setting the bar high and early)
https://www.youtube.com/watch?v=5R8EpAv4miA (nice use of maj7 chord)
https://www.youtube.com/watch?v=pJh9OLlXenM (the whole album is core psychedelia) https://www.youtube.com/watch?v=F-kVFfKezVo
https://www.youtube.com/watch?v=EUY2kJE0AZE (classic Alice-drug references)
One thing that comes up very often in psychedelic music is hypnagogic techniques, especially drones and repetition. These often make it very challenging to compose harmonic complexity, but they open up doors for experimentation in tone and rhythm. Quite a lot of music that takes inspiration from African traditional folk music (like Boredoms or Animal Collective, or some [ambient] dub) use polyrhythm and drum/tape loops to invite the listener to a meditative state by not overtly challenging the listener. It encourages a different mode of listening to music that is far more subjective and emotional, focussing on creating an environment for the individual to have a personal experience, which is very psychedelic in nature.
However, psychedelic music encompasses so many traditions and styles because of people’s varied reactions to psychedelia and what it means to them. A lot of “psych rock” that’s basically garage rock doesn’t even sound very reminiscent of the psychedelic experience, but it influenced many of the makers, so there are a lot of different ways to make psychedelic music, and it can mean a lot of different things to different people.
I’d recommend thinking about why you want your music to be psychedelic- is it because you want it to be similar to your influences, that are referred to as psychedelic? Is it to describe and emulate the psychedelic state or express psychedelic experiences you’ve had? Do you want to discuss topics that often come up on psychedelics, more complex, multi-aspect topics facilitated by the easy-going logic the drugs or experiences give? Whatever the answer, honing in to what exactly you mean by psychedelia should make it easier to find the influences you want to learn from and understand what about your writing you want to be psychedelic!