I love San Sebastián so much we never went home after our holiday 7 years ago 
It has become quite busy in the summer on he beaches but it is a wonderful place.
Are you coming with kids? Will you have your own transport?
There are a couple of beautiful walks along part of the Camino De Santiago. If you take the tube train out to the next village (pasaia to the north or Orio to the south) you can walk back to the city over the coastal mountains which are absolutely breathtaking. Pasaia in particular is so incredibly beautiful and has a little boat you can take across the harbour.
I have far too many food recommendations but if you want a three Michelin star you should book now. I love mirador de ulia which is “only” one Michelin star but the views...!!!
You also don’t say when you are coming but there are various incredible festivals on. A jazz festival with a huge beach stage and Jamie Cullun this year on in July, big fiesta with fireworks over he bay every night in August, film festival in September etc.
It’s nice to take the boat out to Santa Clara island (or swim there if you’re adventurous!) for a day.
If you have kids there is a funicular up a mountain to a little quaint amusement park.
Loads and loads of gorgeous parks to relax in.
If you are able to travel around a bit there is some incredible landscape, a lot of it used in game of thrones as it’s so otherworldly. San Juan de gaztelugatxe near Bilbao for example plays dragonstone, and zumaia plays the dragonstone beach. There are also mystical caves where witches used to meet (like the basque Salem!) inland a bit, prehistoric caves etc. The railway up to La Rhune, the first pyrenee on the border is one of the most beautiful things I’ve seen, you circle up the mountain teetering over huge drops while giant eagles circle overhead and wild basque ponies graze beneath you.
In the city don’t miss pintxos in the old part. They all get far too busy in the summer with tourists though so you can avoid that by going to another neighbourhood.
I don’t really have advice as to where to stay as I live here but it is really expensive as it’s a fancy city so there isn’t really budget accommodation as such. Some people stay along the coast in getaria for example and get the train in every day.