I tried to memorize them but found I still have to write them out.
Easiest way for me at this early point is to start with the C major scale, take the 5th note and start a new scale (so G).
The 7th note of that scale is F, which is now sharp. The 5th in the G scale is D, so you keep F#, take the 7th note which is C (now C#). The 5th in the D scale is A, and so on. G has F#, D has F# and C#, A has F#, C#, and G#, and on it goes.
C D E F G A B C # of sharps
G A B C D E F# G 1
D E F# G A B C# D 2
Circle of 4ths is different, the 4th note becomes the flatted note, and you start the scale with that. Start with C, 4th is F, F scale will have B as Bb, start the next scale with Bb, 4th note is E, now Eb. You notice when you write them all out, they are exactly the opposite of the sharp keys and scales.
C D E F G A B C # of flats
F G A Bb C D E F 1
Bb C D Eb F G A Bb 2