Anything I haven't listed are ones I haven't used enough to say anything about. And I haven't had that much experience with any EDO except 12, so take this with a grain of salt.
5edo - equipentatonic; smallest viable EDO in my opinion and good one to start playing with
7edo - equidiatonic, or if you prefer, "discount diatonic"
8edo - Eight! Eight! Eight! The number of the sed-user!
12edo - 2,1-diatonic. Standard tuning in Western music; has the best support with instruments and software. P5 is great; m3 and M3 are relatively extreme. Highly composite number of steps, so there are a lot of symmetrical chords and scales, and there are only two intervals (plus inversions) that circle around all the notes.
13edo - 3,−1-diatonic (🐹). extremely unlike 12edo, and I haven't played with this at all because it's awkward to notate with 5ths notation (and I prefer to work in Western notation and am afraid of reading piano rolls), but even this can sound good! See this or this example for what it can do.
15edo - 5edo with thrice the notes to approximate 5-intervals as well. Somewhat awkward to notate with 5ths notation. Lacks a good 9/8, making it sound somewhat unusual melodically. Not really a fan of the n2s, which are the building block of porcupine, but MODMOS scales based on it look interesting.
17edo - 3,1-diatonic. Has even more extreme thirds than 12edo. The m3 is closer to 7/6 than 6/5, making minor chords more consonant than major. Because of this, it's suitable for minor pieces. Would recommend either this or 19edo for beginners.
19edo - 3,2-diatonic. Relatively washed-out feel, which might be good or bad depending on the situation. The fifth is tempered quite a lot here. 2\19 isn't so effective as a leading tone, and 1\19 is verging on "too small" for me. Okay septimal intervals, but not as good as 31edo. That being said, it's a relatively small meantone EDO, and semaphore is interesting. Despite its flaws, I'd recommend either this or 17edo for beginners.
22edo - 4,1-diatonic. Relatively small EDO with good 3s and 5s. (Or at least one with a different relationship between them.) 11s are good, with 7s slightly more off. Haven't played much with this EDO because of the syntonic comma.
24edo - 4,2-diatonic. Really good 11, and the 7 is at least better than 19edo. Still suffers (or not, YMMV) from being 12edo but with twice the notes. I don't recommend this one for beginners because it might give you the misconception that microtonality is just about tacking on more intervals onto 12edo.
31edo - 5,3-diatonic. Good all-around meantone temperament, if a little big. It has 11-limit harmony, and it also lends itself to quarter-tone notation. Aside from meantone, it does A-team, mohajira, and orwell, among others. Personally, I'd be reluctant to go bigger than this; 31edo is already verging on "too big".
53edo - 9,4-diatonic. Pretty close to JI, I guess.