You can't go from Sintra to any of those places, you have to pick the train/bus on Lisbon.
You don't have time to check all those places in a single day, even if you rent a car.
From Sintra you can do a daytour to Mafra by public bus. Trip takes around 1h one way. Time table see here:mafrense.pt/XlsHtml/mafra_%2...
Obidos and Batalha in one day will be difficult with public transport. I support the idea of a car. Although the parking situation in Sintra is tight. On the other hand you could go on top of the Sierra da Sintra and to the former church on Peninha, where no bus goes and you have a wonderful view over Cabo da Roca, Guicho Beach, the Atlantic and even to Lisbon.
It's not easy to get between Obidos and Batalha by bus, let alone get to either from Sintra.
Mafra and the Palace is definitely worth a day trip and easy. Be aware that the bus to Mafra leaves from the big bus station at the Portela de Sintra railway station on the eastern edge of town and not from Sintra town centre itself.
And check carefully which days the buses run - the timetable information is only in Portguese.
I must say that those 3 spots were very well chosen. But unfortunately for you, it's impossible visit them in one day by public transportation. Even with a car it's tight. You have to leave Sintra at 8h30, to start visiting de Convento de Mafra early in order to arrive at Óbidos at 13h. At 15h you should leave Óbidos, to arrive at Batalha around 16h. It's quite tight...
you already had some great advice, I will go in another direction, if you have a little time when you will be on your way from Lisbon to Sintra you can spend two-three hours in Queluz, the palace is very nice and the tile fountain in the garden is awesome!