Definitely go into the city.

Leonardo express train is easy enough to use -- good signage from the airport. But then you'd have to take a cab from Termini to your hotel.

An airport cab direct to your hotel is the easiest option. Some hotels will arrange an airport pickup if you ask.

As others have said, don't stay near Termini. Lots of options in the centeral old city.

Satruday night sights/walk could include Corso, Piazza Navona, Piazza Popolo, Spanish Steps, Trevi Foundation.

Borghese Gallery tickets can be purchased in advance.