We've done Majorca on a really tight budget (stayed at a friend's place) with 5 people incl 3 kids similar ages.
It can be as cheap as you can, or as expensive.
UK cereals can be very expensive (and the milk can be wierd) so it might be worth buying bread and having nutella/jam/cheese and meat for a more substantial breakfast which means lunch can be lighter and cheaper.
Deffo go for picnic lunch if you can. Freeze half-filled bottles of water to use as ice packs (then top up with water). This can be as fancy as you like, or buy a lighter lunch.
Self-catering teas again, as basic or fancy as you like - can you bbq? When on a tight budget we ate out only a few times; local beers/wine/spirits are much cheaper.
Eating out for dinner I'd budget min €20 for an adult main, €12 for kids, plus starters/dessert/drinks. Up to megabucks for something fancier.
We're not hiring a car this year as we know the routine for busses and trains. But when we have, it's been cheaper to book in the resort (make sure you take driving licences and have a credit card).
Finally, if possible try and find a big supermarket - Lidl or Carrefour are MUCH cheaper than corner shops. You can buy in bulk (pastries, pasta, sauces, veg/fruit/salads) and still have a much wider choice of groceries than from the spar!