Why such tiny words matter
Two-letter words aren't about scoring 4 points for AT. They're about parallel plays: laying a word alongside one already on the board so that every pair of touching tiles makes a valid two-letter word. One six-tile play can score the main word plus five two-letter crosswords in the same turn. That's how experienced players pull 30 or more points out of ugly racks. The list is also your escape hatch. When the board is closed and your rack is junk, knowing that ZA, QI and JO are real words turns a dead turn into 20 points on a premium square.
The list, grouped so it sticks
About a hundred two-letter words are valid in the ENABLE dictionary this site uses. Tournament lists differ by a few entries. Don't learn them alphabetically. Group them by what they do for you.
Ones you already know: AN, AS, AT, BE, BY, DO, GO, HE, IF, IN, IS, IT, ME, MY, NO, OF, ON, OR, SO, TO, UP, US, WE. Free points, nothing to study.
The power tiles, learn these first: QI (life force, in Chinese philosophy), ZA (slang for pizza), XI and XU (a Greek letter and a Vietnamese coin), JO (Scots for sweetheart), KA (an Egyptian spirit). These six turn the most expensive tiles in the bag into easy plays. QI on a double-letter square is 21 points for two tiles.
Vowel dumps for a drowning rack: AA (rough lava), AE (one), AI (a sloth), OE (a whirlwind), OI. Also AB, AD, AG, AH, AL, AM, AR, AW, AX, AY.
The E column: ED, EF, EH, EL, EM, EN, ER, ES, ET, EX. Mostly the names of letters. EF is "F," and EM and EN are printers' measures. Learn them as a chant.
The odd ones that win arguments: BA (Egyptian soul), BO (a friend), DE (of, from), FA and TI (music notes, along with DO, RE, MI, LA, SO), HM and MM (yes, really), LI (a Chinese unit of distance), MU, NU, PI (Greek letters), NA, NE, OD, OM, OP, OS, OW, OX, OY, PA, PE, SH, SI, TA, UH, UM, UN, UT, WO, YA, YE, YO.
How to actually memorize them
Don't study the list. Play it. A few things that work. Own one starting letter per day ("A day": AA AB AD AE AG AH AI AL AM AN AR AS AT AW AX AY). Learn the meanings, because a word with a story sticks: a sloth, lava, pizza, a sweetheart. And drill with the tools. Type any two tiles into the Scrabble finder and see whether they're valid. A wrong guess you had to check stays with you far longer than a list you skimmed.
Using them at the board
Once the list is in your head, the board looks different. Every edge of every word is now a place to score. Before you play the obvious spot, check whether the same word laid parallel to a neighbor scores double or triple through the two-letter crosses. It works on defense too. When you know the two-letter words you can see which openings you're leaving. Sometimes the best play is the one that keeps a vowel away from a triple-letter square, because you know exactly what QI would do there.
Related: Q without U words for the trickiest tile, and the Scrabble word finder to check any rack. Most of this list works in Words With Friends too, with a few differences. The in-game test play is the referee.