Julie, I think tomas cook & jet2 are your best bet from Newcastle. We have had family & friends visit us this summer, and used both of them. No problems. I booked my daughters flight with t/cook, and it wasn't bad for price, but I did it as far in advance as I could, and I looked for the cheaper dates. (her hols were flexible).
There are lots of apartments which are quiet, it all depends how far from the beach you want to be. When we took our kids when they were younger, we stayed at Tria Adelphia. They have a big pool, and some 2 bedroom apartments, and a maisonette, which easily take 4 people.
They have a bar and do food in the evenings, and all day, but we always found that our kids didn't sleep too early while on holiday, but the bar doesn't get too noisy these days anyway. (the apartments are far enough away from the bar, that you don't really hear any noise.)
I'm sure there will be many more recommendations from other forumers too.
Good luck getting it all sorted....