Me and the kids have done two cycle tours in Northumberland - it is an incredibly beautiful county, well worth a visit. We've done Hadrian's Wall cycleway (from Carlisle to Newcastle) and Coasts and Castles from Newcastle round to Edinburgh.
If you're on limited budget, have you looked at youth hostels? They are fantastic! Nearly always have cooking facilities, so you can save a bit of money on meals. Sometimes cheaper than a Premier Inn, sometimes more expensive - but loads more facilities and much more sociable fun.
Hadrian's wall: you could stay at YHA The Sill. Train from Manchester to Carlisle to Bardon Mill then walk up. They do breakfasts and dinners. On our rest day at The Sill, we walked along Hadrian's Wall to Housesteads Roman fort (past sycamore gap - the tree was still there when we went!) and then got the bus back. The bus service along the wall is pretty good. Vindolanda is also walking distance away, and well worth a visit.
Coastal: Alnwick hostel is super lovely. When we last stayed there, booking was a bit odd and we ended up doing it independently. Barter Books is well worth a visit! We didn't have time to do Alnwick castle, but did do the gardens next door which were fun. We were personally not wowed with Bamburgh castle; it was expensive and closed 30 mins earlier than advertised. But we were a bit castle-d out by the time we got there, so maybe we were just a bit jaded! Our faves were the little castles but of the bigger ones, we enjoyed Dunstanburgh most. The (independent) hostel at Dunbar (just over the border but good train service) was the most incredible hostel we've ever stayed in - I really didn't want to go home...
Avoid the old YHA Newcastle - I think it's no longer a YHA, but can't imagine it's any better. It was a real dive when we stayed there in 2022, so we stayed in a premier inn in Whitley Bay in 2023 instead.
For a short trip from Manchester, I reckon Hadrian's wall is probably a better bet, otherwise you'll spend all your time on travel (although that will be beautiful too). One thing that surprised me a bit was how few train stations there are on the coast - the east coast mainline rushes up and down at top speed but doesn't actually stop much. Whereas the Carlisle-Newcastle line is a little local line with lots of stations and a train once an hour, and the bus service along the wall was good.
Hope you have a good trip! Northumberland is so beautiful. I reckon our coastal trip was more spectacular for landscapes, but both were good.
Edited for spellings.