There a many options to get to the Horse track.
You can take the bus 1A towards Klampenborg, runs every 20 minute, and walk 5 minutes, or take the S-train to Charlottenlund a have a nice 15 minutes walk through the forest.
Or, do as the Danes bike. Only 30 minutes from the city Centre.
If you bike, you will pass through Hellerup, one of the wealthy areas in Copenhagen.