I'm assuming a package might be best if you have kids, but if buying online check you're actually buying a package for maximum protection rather than a series of separate holiday components; and use a credit card for added protection.
You'll probably already know the cheap travel websites (expedia, kayak, teletextholidays, travelsupermarket, travelrepublic, icelolly...) some of which let you search either by 'any destination', or by broad geographic areas.
I'd start with Spain where prices have fallen and look at popular family destinations; for example, if you select 'Balearic bestsellers, or 'Benidorm bestsellers' on teletextholidays.co.uk you'll already get all-inclusive offers under £300 per person per week in hotels or resorts which you can then cross-check against TripAdvisor reviews. Not all these popular resorts are as tacky as you'd imagine- we love Mallorca, although we tend to go off season to smaller places. We've also taken the grandaughters to northern Spain (via Bilbao or Santander) in August and had good value.
Its a difficult call as regards when to buy. I suspect it's far too early to book for next August and that prices will fall in 2103. You could also take a risk and hang on for last minute deals with many of the sites above, or with the big holiday operators themselves (Thompson for example are still showing silly prices of £600+ pppweek)
Alternatively, if you don't mind arranging your own accommodation and transfers from an airport as Steve suggests, you could try my favourite cheap flights scanner http://flightchecker.moneysavingexpert.com
This lets you search flights from UK airports either by destination or an 'I'll go anywhere' option within specified countries and lets you input a wide date range too. August'13 flights are already showing under £80 pp return to, say, northern Spain and will probably come further down in February in our experience.
The problem with these cheap airline scanners is that while Easyjet have already loaded many summer 2013, other cheap airlines like Ryanair haven't. And also, on reflection, its probably not feasible to travel hand luggage only with kids? We can cope with one small bag each even for 2-3 week holidays but if you check bags in on Ryanair, you'll find they're so expensive that they can double your fares!
The moneysavingexpert site also has great general guides and web-links about low cost holidaying - http://www.moneysavingexpert.com/travel/
Good luck- you'll get there!