First trip to Italy!!!

About train schedules, that is mostly for Regionali. Freccia trains, which are the high speed trains between major cities are already up and available for purchase for the most part. Regionali schedules usually don't change much, so you can look at the same day and time for trains in the next week to get an idea. Regionali prices are fixed (and partly subsidized by the government, thus the delay for negotiations) and there is no advantage to buying in advance.

Wonderful Itinerary. From what I have read the regional trains run from Matera to Bari several times a day and I believe it is regional as well to Polignano a Mare. Not too many people on the site have been to Polignano a Mare. In my research on Matera I looked at the map to Alberobello and the Truilli houses and then found Polignano a Mare on the map and thought it was unique and certainly the restaurant Grotto Palazese is a must do for us. Enjoy your well researched trip and I am sure you will be back again. This is our 7th trip to Italy and have never tired of the people, scenery, food and wine.

"We are starting our trip on 20 Jul 18 and will be returning home on the 4th of July."

Could you clarify that? Are you starting your trip on 20 June?

As Sam indicated in the previous reply, many of the schedules for the high speed trains are already posted, however there may still be a few that will be uploaded over the next few weeks.

For a trip from Bari to Rome, I'd suggest using one of the direct (no changes) Freccia trains, as some of the others have an inordinately long travel time due to long layover times at intermediate stations (ie: Ancona). Travel time for the direct trains is 4H:06M, and there are departures 07:14 and 13:14 (among others). The 07:14 departure appears to be one of the cheapest. Which day are you planning to make that trip?

For the second part of your question you'll probably find "authentic food" at many restaurants, whether in "tourist areas" or otherwise. I'd suggest asking the staff at your hotel for suggestions on "authentic" local restaurants. I've always received good tips from hotel staff. One point to note is that if you venture into local neighbourhoods far from the tourist areas, the restaurant staff may have limited ability to function in English.

Thanks for the replies, yes it is June 20th that we are leaving and we are traveling from Bari to Rome on Sunday, July 1st.

The later you eat the more local the crowd becomes. Tourists tend to eat earlier than the Italians...
My rule of thumb to get "authentic food"...I never eat where they offer hamburgers on the menu. Works like a charm.

For a better dining experience, eschew anywhere that has a “barker” out front enticing you to enter. That is, IMO, the kiss of death. Good local restaurants do not need to beckon. In fact, a thoughtful traveler should make a reservation as it is a sign of respect and you will get a better table. As mentioned above, dine late, after 8pm. Only tourists eat before that. In summer, we often dine at 9pm in Italy, when it is cool enough to enjoy our meal.

Also, stay away from restaurants that have large "picture menus" in front, featuring unappetizing pictures of their menu choices.

Gee, swulbrandt, I guess you didn't get the word. There is a new law in Italy, and people-watching is no longer allowed. But maybe you can still get in a few peeks without getting caught! Enjoy your wonderful and somewhat unusual itinerary in Italy!

Choosing a restaurant - don't be put off by how it looks, some great food is served in very simple settings. Ask to see the menu, if it's in more than two languages (most places have English translations), it's for tourists. If food is displayed in the window, keep walking.

To avoid burgers you would have to bypass 85% of the restaurants in France.

From Forbes: "But it's not just as a favorite fast-food that the burger has surpassed other French food traditions. Hamburgers of all varieties, from those of the fast-food chains to sophisticated posh creations by famous chefs, appear on the menus of 85% of France's restaurants."

It appears burgers have become quite trendy in a number of Italian locales but hopefully you wouldn't have to avoid 85% of the restaurants.

https://www.forbes.com/sites/ceciliarodriguez/2018/03/20/france-taken-by-hamburger-hysteria-with-1-5-billion-swallowed-per-year/#241efb3253b3

The French certainly have a burger thing going...not so much in Italy. I also avoid burgers in France but not places that serve them. That would take just about every Bistro out of play.
But in Italy, when they are offered, by and large it's for tourists.
All things considered, we should not give the impression that you meed to dine in exclusive restaurants to enjoy a decent meal. Almost any place you walk into will serve you a better meal than 99% of the eyetalian restaurants here in the US.... the food is good... except one place where they broke the sauce in a carbonara (sad....but not an Italian in the kitchen I learned later when I was washing my hands) we did not have a single bad meal in 3 weeks of travel...there were some that were spectacular but none that were bad...not one. If it looks interesting, if there are folks there, give it a shot.
One final point to consider....most good restaurants fill with reservations. We sat near a window in a Trestavere restaurant where I could see folks checking out the menu. They would come in and ask for a table and get turned away even though the place was not full...while we ate the entire place filled...all of the tables (including ours) had been reserved in advance. Sometimes soem advanced planning is necessary.