What’s the best way to keep track of my bets on horse racing?

Great question, and the fact that you’re asking it is a good sign that you’re serious about profiting on horse racing.
Keeping detailed records on your wagers is very important if you want to continue to improve your betting and handicapping.
I recommend creating a spreadsheet that reflects your results by month as well as the returns you’ve gotten on the various types of bets you make. In my case, I use Excel to record my overall returns, with separate tabs for win, place and show bets, exactas, trifectas, Pick 3s, Pick 4s, Pick 5s and Pick 6s. That makes it easy to see which wagers are dragging down my overall ROI.
If you use an advance deposit wagering (ADW) provider and exclusively make your bets online, you may be able to accomplish the same thing with much less work, simply by accessing and sorting your betting records.