Many beginners imagine they must memorise a great deal before they may pray — and so they postpone praying. In reality only one surah is obligatory.
The obligatory one: Al-Fatiha
Seven verses, recited in every unit of prayer (rak‘ah). Whoever knows it can pray validly. It is therefore, without exception, the first thing to learn.
Then three short surahs
Al-Ikhlas (4 verses), Al-Falaq (5) and An-Nas (6). They are recited after Al-Fatiha in the first two units, and they let you vary your prayer instead of repeating the same thing.
They are the shortest in the Quran: a consistent adult learns them in three to four weeks.
The formulas for bowing and prostration
These are not from the Quran: they are short formulas learned in two days. “Subhana rabbiya-l-‘Azeem” while bowing, “Subhana rabbiya-l-A‘la” while prostrating, and the tashahhud at the end.
How long in total?
Al-Fatiha: two to four weeks starting from zero. The three surahs: about a month. In other words you are praying a complete, varied prayer within roughly two months.
And before you have memorised it all?
Prayer is never postponed while waiting. Whoever does not yet know the surahs recites what they can, or keeps to the formulas of glorification. An imperfect prayer is far better than a postponed one.