Just addressing the top line. It's in 4/4 and you count 1-2-3-4- one number on each beat. The & in between each number splits each beat into two half beats _ quavers or eighth notes (as then, there are 8 in each bar).
At the beginning of the bar, the kick drum plays with the first hi-hat hit. Snare comes with the third hi-hat hit, on beat two, then on 3 and the & after 3 the kick plays with each hi-hat hit. On the count 4, the snare gets another hit.
It really doesn't matter how slowly this is played initially, so count 1 & 2 & 3 & 4 & while hitting the hi-hat each time. Then add the kick at the appropriate points, and when that's all automatic, fit the snare in between. Speed up only when you don't need to count. It's a very well used and popular 'beat' in pop. To me it looks like a single bar, where often 8 hi-hat hits feature.
The lower two pics are indecipherable, but 'let' may be something in relation to cymbals, which can 'let ring', as opposed to most drum sounds, which are if very short duration.