Llovía
(the preterit is called for in this case - although many narrations about something in the past start out in the imperfect, this past event ties to the next event - both had an end in the past)
(Llover) mucho ese día y, por eso,
decidí
(action had an end in the past)
(decidir) buscar un taxi. Claro,
habían
(use the singular form, but you chose the correct tense)
(haber) muchas otras personas que también
buscé
(should still be relating what was happening (also, wrong person and number (as an aside, you've misspelled
busqué)))
(buscar) taxis por el mal tiempo que
hació
(should still be relating what was happening)
(hacer). Por fin, (yo)
encontré
(action had an end in the past)
(encontrar) un taxi y subí. Le dije al taxista adónde
quise
(the preterit form of
querer means 'tried' (no longer want))
(querer) ir cuando de repente
recordé
(action had an end)
(recordar) que mi bolsa
estuve
(describe an ongoing action)
(estar) en una de las tiendas de la plaza. Bajé del taxi y
fui
(ir) corriendo hacia el mercado. Cuando
llegé
(wrong spelling of
llegué, but you chose the correct tense)
(llegar) al mercado,
vi
(ver) mi bolsa. Yo
estuve
(describe a state that was ongoing)
(estar) muy contenta.