Just a small notebook with a pen, or the text/notes app on your phone.
I record every workout: each lift, weight(s) used, sets and reps, and anything special (drop sets, for example). But I don't go crazy on details. It helps me progress over time. If I pressed (overhead), say 135 lbs, 3 sets times 5 reps, then I'll know to aim for at least that much next time, and to eventually progress to a weight higher than 135 and/or more reps than 3x5.
At the very least, record your basic lifts. Your isolation movements matter less, but I record those too.