I spend around QAR1800 for a family of 4 but with two small kids under 5. We are not vegetarians, we do buy branded food like Kelloggs instead of generic and the groceries I mentioned are just for food costs for home cooked food. Our staple is rice.
Hope that helps with your groceries' estimation.

Yes re the internet, especially if you want a decent speed for downloading or uploading. Please err on the high side for your budget (as per Mary Catherine's post on expenses) because monthly expenses seem to go up quite a bit every year.