We travelled from Mumbai to Goa via Pune, Kolhapur, Amboli, Sawantwadi. The route is pretty good and overall journey was around 600 km. If you have your private car then it should not take you more than 10 hours with some short intervals.
Suppose, if you start early in the morning at around 7 am from Mumbai, then you should reach Goa by early evening. So, I don't think there would be any need for a long halt. But still if you plan for one, I will suggest you to stay at Kolhapur, as there are many good hotels and restaurants nearby. In your leisure time you can plan a visit to a temple or a lake in the evening.
You can again start your journey for Goa via Nipani, Sankeshwar, Gadhinglaj , Ajara, Amboli, Sawantwadi and then Goa. Most part of your journey would be via NH-4 and then AH-47 from Kolhapur, followed by some small but good roads.
1) Try not to start your journey at night, as driving at night through some of the ghats can be very challenging.
2) A person who is totally vegetarian can't be too choosy as the options would be limited. Non-vegetarian person would really love the food though.
3) Make sure that your car is up to date as the journey is long and you would not want to be stranded in the middle of nowhere.