The sanctuary (or park if you will) that is closest and easiest to reach from Pune. You drive yourself around or could potentially also walk if you feel like it. Mayureshwar is best explored through a combination of the two. Drive around slowly looking for interesting wildlife and then potentially getting out of the car to position yourself as good as possible when you spot something.
Chinkara (Gazella bennettii)
The main target species to see in the reserve is the Chinkara, a species of Indian gazelles.
Birds
Of course there is more to see in Mayureshwar than the Chinkara. It's also known for it's steppe birds including the beautiful steppe eagles.