Le Marche

I haven't visited that part of Italy, but if you have an itinerary sketched out, you can use Rome2Rio.com to find out which travel legs are possible by train or by bus. It's important, though, not to trust the more specific information provided on that website. The frequencies, travel times and fares are so often wildly off-base as to be more dangerous than useful. But if you keep drilling down, you'll eventually find the name of the company providing the bus service; just go to its website to check on the schedule.

Trains run along the coast, but not inland. I have only visited with a car, for instance to Urbino. Bus service designed for locals might be sparse, e.g., might primarily serve school or business commuters who are possibly going the opposite direction to you (from a small town to a larger one in the morning).

I've done it and it's very time consuming bc it's not well connected. For trains, you need to stop in Bologna and then take a regional train to Pesaro, a beach town. From there you can pick up a bus to Urbino, Urbania or other parts of the region. The buses are nice and comfortable. Don't count on people speaking alot of English there either so bring your phrasebook.

I'm not really answering your question, but we rented a house there for a week in 2008. We had a car, and it was the easiest driving I've ever encountered in Italy. (I've driven a lot in Tuscany, Umbria, and Sicily.) Roads were mostly two-lanes and not heavily trafficked. People were friendly, and in pre-season May, the beaches were empty. Some of the best (and cheapest) seafood and Italian food I've ever found. Many restaurants were very limited in English, so polish up your Italian or prepare to be adventurous. There were not a lot of big tourist sights, but it was wonderful to find places where we were the only visitors. We went to market days, the beach, some kind of military festival reenactment about which we were completely clueless, and many lovely, small family restaurants. That was twelve years ago, but it was delightfully untouristy. Thinking about it makes me want to go back soon!

The problem of Le Marche as region is that is mostly mountains and hills, so very difficult create an important railway system. Main (biggest) cities are concentrated on the coast or not very far, so along the years has been created an important railway near the coast, but few and secondary railways inland.
The buses are designed in Italy at a Province level, so often not so suitable for tourists as timetable. As told before thought mainly for students and commuters.
The final answer is simple: a car in much better. Prepare yourself to drive without so much traffic (except cities), but on winding roads climbing up and down hills. Be careful in historical center, even if is a small town or village, about zones where you cannot drive.

Thank you all for responses. Those are what i was expecting, but figured i'd see if anyone did public transport successfully. We don't mind driving, but would prefer not, but thank you for confirming our expectations. Much appreciated.

David we spent 5 nights in Pesaro in 2017 without a car. We were there off-season, early October, and had a 5-star oceanfront modern hotel for half-price. There were few other tourists. It all suited us fine however, we were very limited by not having a car. Day tripping to Ravenna was easy, but Urbino was by local bus and a longish trip as a result. We had little access to other areas as the bus connections were spotty and led to long trips. So in 4 full days we saw Urbino, Ravenna, and waaayyyy too much of Pesaro as there is not much there. Buses on Sundays are infrequent. But it was a place to relax in 70 degree sunshine in October when Italians think the beach is too cold.

We have done 2 trips the Le Marche. The first time was from the north (from Ravenna) by train. We then went to Pesaro, with a bus trip to Urbino. Then back to Pesaro and a trains to Macerata and Ascoli Piceno. We eventually took a train from Ancona west to Foligno then on to Rome. The trains going were more local and slow, but fine. Our second trip, we took a rental car from Rome towards Citinova Marche. We focused mainly on the smaller mountain towns. We enjoyed both trips. I would rent a car only if you have a chunk of time. We found travel in Le Marche to be very pleasant, but low key.

You could go up and down the coast by train, and get a few places by bus, but it would not be worth the hassle without many weeks to do so and if you had a major problem with driving. The driving was super easy.

Thank you.