Of course you should. You want to know what your PRs are. You want to hold yourself accountable. There have been times when it was tempting to only do two sets rather than three sets of a particularly heinous exercise (think reverse hyper, or weighted GHD sit-ups), but then I'd see the the two set numbers looking at me, mocking me for being a sissy. FINE! I'll do my third set! GEEZ!
If you do your own programming, you'll want a record of what you did last week so you know if you need to bump up another 2 kilos this week, or change one movement in for another.
They are also helpful for tracking notes on new techniques, or how a workout generally felt to you and how your body feels day to day. Tracking these along side your numbers can help you elucidate performance problems outside the hours you spend in the gym.
What are the best ways to track?
I'm a big fan of the simple paper and pen method. I'll get a fun journal, a bunch of colored pens, and just write down everything I do. Take notes on if something felt off. Note new technical cues that made a difference. I'll usually pick either the inside cover of the front or back to track 1 rep max for lifts, crossing out and adding with each new PR. It gets messy, but it works. I usually also write each day in a difference color, so I can easily put several days on one double page spread and see where one day ends and the next starts.
Some people would rather have electronic tracking versions.
Most of the tracking things I found are for the iPhone, of which I do not possess. But here is a good post about then anyway:
11 iPhone apps for tracking your workouts
Then there are a lot of templates on Google Drive to use if you like something you can access on your phone, tablet, or computer:
Search list of Google Drive Templates
For a while, when I was also tracking my food (which I should do again), I used FitDay to track my exercises and moods as well. Mostly, I turned it into this perverse game of "can I eat enough calories to make the program think I'm going to get fat, and still maintain or lose weight"?
The moral of the story is, there are a lot of ways to keep track of your progress, but keeping track is the main point!