Day trip to South Shields. Get the ferry from North Shields, it takes around five minutes to cross and only costs a couple of quid but kids love it. My DC reacted like they were on a Disneyland cruise 😂 When you get to South Shields, cross straight over the road and there's a big building on your right called The Word. They have free activities and exhibitions on all summer. Walk down into the town and there is a free museum, think it's called South Shields Museum, about the history of the area with various things to look at. Head towards the beach and there is an amazing park on Bents Road with play areas, a sandpit, a duck pond, a huge playing field, ice cream shop/cafe, and a miniature railway. There is a funfair and arcades across the road which can be quite pricey but if you check their FB page before you go they often have offers on like £25 worth of ride tickets for £15. For reference, £25 worth of tickets gets my three older DC around 4-6 rides each depending on which ones they want to go on (the little kid rides are fewer tickets per go than the bigger rides).
Discovery Museum and Great North Museum, both have summer programmes on their websites of free and cheap activities plus the museums themselves are free.
If you can travel then Ridley Park in Blyth has a free splash pad with water fountains. Between the water, the play areas, and the lawn games outside the safe we easily spend an entire day here. Take a picnic for lunch and all it will cost you is the price of an ice cream.
Just along from Ridley Park is Blyth Beach. They have beach huts you can hire for £16 a day, there is an ice cream parlour, chip shop, park, and a WW2 battery as well as a lovely beach to play on.
Tynemouth Park is good for a day out too. For £10 you get a game of mini golf, a go around the maze, and all day access to the inflatables (slides, bouncy castles, climbing walls). There is a duck pond with boats and a play area too, plus the beach across the road.
The big movie screen is going to be on at old Eldon Square/hippy green showing films in the open air for free. My DC love sitting with a bag of sweets on a deckchair watching cartoons.
Buy a cheap paddling pool or sandpit for the garden if you can, free entertainment at home then too.
Morpeth park is lovely, it has a paddling pool and nice play area as well as walks along the river where there are ducks, canoes to watch, bridges for pooh sticks, etc. Plessey Woods also.
Look up train tickets, last summer I got all six of us to York for the day for £35 return. There is loads to do in York for free, it's only an hour away from Newcastle, and the kids think they're on an adventure.
Whitley Bay has a small but nice park with sand and a water play area, there are some arcades around the corner with 2p machines and tickets to collect for little prizes, and a nice walk along The Links or a play on the sand.
I don't know if it's still on but Groupon had an offer with half price annual membership to Seven Stories. They also often have offers for half price bowling, half price softplay, and half price climbing.