The most essential day trip from Panaji is Old Goa, the abandoned former Portuguese capital nine kilometres upriver. The complex of 16th- and 17th-century churches and cathedrals — Bom Jesus (with the relics of St Francis Xavier), Sé Cathedral, St Cajetan — is a UNESCO World Heritage Site and one of the most extraordinary colonial-era ensembles anywhere in Asia. Half a day at least.
The North Goa beach belt — Calangute, Baga, Anjuna, and the smaller villages stretching up to Arambol — is twenty to forty minutes by road. Mapusa, the market town inland from the beaches, has Goa's best Friday morning market. South of Panaji, Margao is the bigger commercial town and the gateway to the quieter southern beaches of Colva and Palolem.