all you need to know for 2x2 BLD:
you only need to use the alg RU'R'U'RUR'F'RUR'U'R'FR
the alg swaps ULB to DFR and the U face sticker gets put on the R side of DFR. You use setup moves to bring whatever peice is next to the DFR location but only setup moves that don't move arond the ULB corner. solving goes like. setup moves, alg, undo the setup moves. If the ULB ends up solved, just swap with another corner that isnt solved yet, and continue to solve as normal. Also at the end if any peices are in place just flipped then ULB will be flipped also so just bring flipped peice to UFL and if needed to be CW twisted then do RUR'U' L' URU'R' L then put peice back to location. if needed to be CCW twisted then after bringing to UFL do URU'R' L' RUR'U' L and back to location. To memorize you should memo just the first letter of the colour you are swapping and tap the position where it is going so you remember what peice it is. (example: if start of memo was O and you knew it was DBR then the peice is obviously Blue, orange, yellow)
memo should normally be no more than 6-7 letters and memorize the positions where you swap to.
That's basically it
edit: example solve,
scramble: R' F2 L2 U2 F2 U2 B2 R B R2 U (White U Red F)
first peice in ULB is YOG with Yellow on top so that goes to R of DFR.
so the YOG peice would be in DBL and you need to bring the side that Yellow would be into R of DFR.
so you would do the setup moves, DF' then alg, then undo DF' so thats (FD') then that peice is solved.
the rest of solving would go like this:
next peice: BYR - just do alg
next peice: WBR- setup (F), alg, F'
next peice: ULB is solved, so switch with any peice (DFL) - D, alg, D'
next peice: WOB - setup (RD'), alg, DR'
next peice: GRY - setup (DFD), alg, D'F'D'
next peice: all solved but need to twist UFL so: no moves required so just CCW alg (URU'R' L' RUR'U' L)
memo would be: YBW, new cycle, WG.
Then hopefully don't forget to twist UFL at the end
then solved!
Hope it helps.