Is it possible to visit Old Goa from the beach resorts?

Your question is old but this answer might be useful to other users.

Is it possible to visit Old Goa (for a day trip) from one of the beach resorts? How long would a taxi drive take and how much would it cost?

---> hooray has answered these questions.

Or is it better to spend a night or two in Panaju, and then move to the beach?

----> Instead stay where you are living, take a bike on Rent it would cost you from Rs. 150 to Rs. 600.00 or 700.00 per day depending on bike.

If you take a bike on Rent, few precautions:

1. Take all legal documents of bike from bike owner
2. Keep your license with you
3. Helmet is compulsory.
4. These bikes have black colored number plates with number written in yellow color.
5. As you will be going into Panaji, you will definitely checked by Police.