I have been biking regularly around the city for over a decade. Mostly the west end, downtown, east side, Southside and Olneyville (so not Smith Hill or Mt Pleasant). My advice is to be calm, alert and give yourself space. Assume the worst of drivers. Like if you're close to a car on the road, assume he/she might make a right turn without signaling and almost swipe you. Again, give yourself space and always be prepared. If you expect the worst, you'll minimize your chances of being caught off guard.

The roads can get crowded, especially during rush hour and when there are cars parked too. Don't be afraid to use the middle of the lane for a bit.

Honestly most drivers are fine and I think used to the narrow busy streets of Providence. But you will run into some dodos too.

Providence has also immensely improved their biking infrastructure the last few years although there is a lot to improve too. If you have specific routes you need help with I'd be happy to weigh in!