My trip to the North of England

Have you already visited Hadrian’s Wall? That would be on my radar at Newcastle.

I haven't because I won't have a car. Is visiting via public transit possible from Newcastle?

Presumably you are flying back from Manchester.
No matter how early your flight is there is 24 hour bus service to the airport on two different lines from Piccadilly. Takes about 50 minutes. ( half the time it takes by day).
In July the buses should not be too crowded, but in term time the 43 especially can be rowdy with University students in the night time economy, which doesn't wrap up until after 4am.
I came into the city on a night airport service the other night and was surprised by how busy we got from Rusholme onwards.
No real comments on your itinerary. Sheffield and York are both very easy and short day trips from Manchester.
You clearly know your itinerary and what suits you.

Yes, Hadrians Wall is dead easy from Newcastle. Train to Haltwhistle then the AD122 bus every two hours to all key sites.
Note the summer enhancement of the bus to hourly is not happening this year.

One option that makes for a fabulous day at Hadrians Wall is to take the train to Haltwhistle and have Peter Carney (guide) pick you up there and show you the key sites in the area. Well worth the money.
https://www.hadrianswall-walk.com/

A couple places I have bookmarked to see in Newcastle Upon Tyne are Angel of the North, Beamish, the Living Museum of the North, Newcastle Castle, Jesmond Dene Falls, and Old Mill. How you get to some using public transportation, I don’t know, but it should be pretty simple.

Durham is a lovely day trip (its cathedral is my absolute favourite for its Norman columns and also its strong sense of place and community) but it’s not huge so you will have time to see some of Newcastle on that day as well. The Newcastle Quayside is well worth exploring. The city has some iconic bridges and the Baltic modern art museum (technically in Gateshead as it’s on the other side of the river) is free to enter with great views from the top floor. It also has a very cheap pay what you think cafe.

Grey Street is a beautiful sweep of Georgian architecture heading from the city centre down to the river.

Newcastle is known for its vibrant nightlife. It’s a working class town with a history of living for the weekend. I would highly recommend a drink in a Quayside pub. They’ll all be busy from early evening pretty much any day of the week and the locals love to chat so don’t be shy to strike up a conversation at the bar.

You only have 1 - 1 and a half days in Newcastle so you won’t have time to get to Beamish or anywhere outside of the city. I’d focus on the city centre areas of the Quayside, the Castle (which is Norman) and the city walls of the same period. There is also the Grainger Market, a large Victorian indoor market.

If you wanted to venture further then Tynemouth would be a superb easy trip on the local Metro light rail. It’s a very lovely seaside town with a ruined Abbey, golden sands (it is chilly though!) and plenty of good places to stop to eat and drink. You can walk for miles along the seafront should you wish.

I do figure my day trip to Durham won’t last the full day. I do want to see the cathedral, both on a tour and for a service if possible. I’ll also see the castle, walk over Kingsgate Bridge and up Saddler Street. I’ll have a coffee and pastry at the Cafedral and a pint or two later at one of the real ale pubs for the rest of my day there.

Re Jamie above- transit details are on each attractions website, but briefly Beamish has two buses an hour from Newcastle. Jesmond Dene has 2 close by metro stations, and more citybus services than you could ever need.
Angel of the North has bus route Angel 21 passing it every 10 minutes on the way from Newcastle and Gateshead to Durham.

Too much to do in one trip https://englandsnortheast.co.uk/

https://www.theguardian.com/travel/newcastle+travel

It does also seem that Dale is potentially mis-informed about public transit in rural areas. If he is from many states in the union that is understandable. If he asked the right questions that could potentially be rectified.

Dale, I applaud your idea of staying in Newcastle. I've been there twice, most recently last year when I stayed five nights, I believe, and didn't see nearly everything I wanted to see. I was so taken with this vibrant city and don't know why it is so under the radar for many from the US. I second the recommendation to try to hire Peter Carney for the Hadrian's Wall day trip. He is amazing, a former high school teacher, I believe, and a big history buff with tons of information to share and is an all around lovely person to spend the day with. I see you have a day trip to Durham on your itinerary. We went to York on a day trip and had a great time, so much to see there and walking the wall is unforgettable , although Durham is worth a trip too and a bit quieter in my opinion than York. We also took a day trip to the sea coast. I believe we were able to take the Metro out there as there are a couple of attractive coastal towns that are not far from Newcastle. (I see now that Helen already suggested Tynemouth.)

I'm surprised the you found Newcastle inexpensive. I thought the hotels were overpriced, but perhaps you're planning to the stay in the middle of the week. The city seems to swell with revelers on the weekends. I could give you some recommendations of places to eat. With the help of our motel, we discovered several good eateries in the center, but it looks as though you won't be there long. Have fun. It's a great city and very good transportation hub as you've discovered.