For a nice small resort try Arillas on Corfu.

Here's a link.

http://www.arillas.com/page/

Thank you, I will check it out

Kedi,
I second Arillas as an option. It does have a couple of bars, but the main focus are pleasant tavernas, on the beachfront or back a bit. There are enough to keep you going for a week. The beach is not fantastic, but pleasant enough. In terms of prices to eat out, I'd put it a bit lower than the beach-front at, say, Bitez. It's mainly package tourists, but more apartment-type accommodation and small hotels.
There's also San (Agios) Stephanos (NW) a wee bit round the coast that might be worth a bit of investigating. It's a small family-type resort area, with a large sandy beach, lacking in mad water-sports, it's Blue Flag and has gently shelving water.
Glyfada is a stunning beach (watch the waves), if it's beach you want, but not sure about the accommodation options as I only visited for the day. I went in September and it was pretty quiet.
You might want to have a look around the following site for some ideas on the different "resort" areas and beaches. I find it quite honest and up-front as a starting point.
http://www.agni.gr/guides/resort_guide/resort_guide.asp
I can't swear that you'll meet "no Brits abroad" in the places mentioned; however, you'll find low-key resorts, with some authentic tavernas (you will find chips) and a lack of a too-British feel IMO. If you hire a car, it'll pay you dividends in seeking the places that are going to give you that "stumbled-upon" place for a surprise lunch.
Don't miss Corfu Old Town early evening, it's a treat with old, winding backstreets, little shops and places to pitstop and eat/drink.

Thanks for the information Phoebe, I will check out the places you have mentioned

Hi
Corfu has many resorts great for families, my own recommendation would be Acharavi in the north, very much an all year round Greek village,http://www.corfucastaway.com/acharavi_area.htm or Ag Stefanos on the north west coast, both sandy beaches, quiet, but with good choices of bars and restaurants. http://www.twfcorfu.com/agstefanos.htm
water ports available in both places and a good choice of excursions if you want to get out and about. Acharavi also has a small water park if you feel like a more energetic day out.

Thank you, more ideas to look at

For a nice quiet family resort check out Agios Georgios north http://www.agiosgeorgios.info .
At this beautiful resort there is a huge sandy beach and some water sports with boat trips too (small boats) including trips to nearby islands . The only 'Brits abroad' will be people like me who avoid noisy brash places and drunks of any nationality.
There are some nice quiet tavernas overlooking the beach, just a few small hotels and mostly locally owned apartments.

The beach does look lovely, that is what we are after for the girls. Thanks, I will take a better look into Agios Georgios