We adore Guernsey! We have had lovely short breaks at the Jerbourg. It's a proper old-fashioned (kind of Miss Marple old fashioned, not actually historic) hotel on a cliff top and they're very child friendly. It has an outdoor pool, nice patio area, good food and spacious rooms (although not 'fancy' or upscale as such). Very friendly staff as well. Not sure if it would be expensive for a whole week as we have always had a half board short break deal which was very reasonable. There is a bus stop right outside and you can take the bus all the way around the island. There are two other places where we have eaten and thought they would be nice places to stay, one was on the way down to Fermaine Bay and one was called the Auberge. I think they would both have been more expensive though, the Jerbourg ticked the box of reasonable and very pleasant.
We stayed in a self-catering apartment out near Vazon Bay called Beau Vallon, which was also nice, although there are loads of self-catering places with pools and facilities and things. I'm not sure how handy the bus would be for Beau Vallon but it was on a main road (for Guernsey!) and so I assume it would be fine.
Not stayed in St Peter Port or anything.
I'm v jealous that you are staying on Sark though, we had a gorgeous day there and I wanted to stay, possibly forever.
If you don't have a car I'd stay in St Peter Port. The Old Government House is a lovely hotel but pricey, they do have a cheaper sister hotel called the Duke of Richmond which might be worth investigating. Next time we go we might stay there. There's also a Best Western that gets good reviews.
Have dinner in Christies while you are there, great food and if you go early you can get the early evening menu and it's fantastic value for money.