Pesaro Italy as a Base

We spent 4 nights there a few years ago and found it a pleasant place to stay but to do you will want to get around a bit. The seafront is peaceful and by September most of the crowds will be gone. The beach was literally closed on October 1 when we were there, meaning no services but pleasant to walk.

We enjoyed a day in Urbino (by bus) and another in Ravenna, by train. San Marino is a possibility and we thought of going but decided on a no transit day as the following day was going to be an almost all day en route to our next destination. There are other towns along the coast like Senigallia, Ancona, worth exploring. Not sure about efficiency by public transport. A car would make some nice outings more efficient.

Whether it would be more convenient to have a car or not depends on what places in the region you plant to visit. Some major destinations, such as Urbino or Ancona can be easily reached by public transportation. If you want to explore the many beautiful villages in the area or the Frasassi grottoes, then a car is advisable.

Fano is nearby and if you have a few minutes to walk around a quaint small town. You can take the bus to Pergola which I highly recommend.... VERY small but the bronze statues are amazing and the streets are fun to meander around. We also took the bus to Urbino. My friend lives near Pesaro so each time I visit we visit some small towns. A car would certainly make it easier in that area.

We stayed there in early Sept a few years ago. Nice town with a pedestrian-only core and a beautiful stretch of coast. The Adriatic side is where many Italians go for vacation and you'll see fewer English-speaking tourists. However vacation time for many Italians is July and August so most (but not all) of the beach spots where you can rent an umbrella and beach chairs were closed when we were there even though the weather and water were wonderful.

If you're spending a full week there I'd recommend a car unless you just want to relax on the beach for the week. Be sure to understand the ZTL signs and carefully obey all posted speed limits. If you have an interest in pottery then Faenza is famous for it and about an hour drive away. Urbino is easy to reach but we enjoyed as well relying on serendipity. You're driving around and spot a town, just pull in and take a walk around!

Ditto--it depends on where you want to go.
Main towns like Urbino will be reachable, and the train line zips along the coast.
We had a car and stayed in Senigallia and Fano, both reachable by train. The park north of Pesaro is gorgeous and you would need a car.