OK, just to get you started with a few ideas.
The most expensive part of any turkish holiday is the flights, especially with a party of 5. Fortunately there are 2 budget carriers. Easyjet, which flies LGW to Dalaman and pegaseus flying Stanstead to Bodrum. The easyjet flights start from c. £100 return. Their summer schedule, taking you up to 24 October will be released this Friday 5th December (according to their website) but the cheap tickets do go fast, so you might have to make some quick decisions if you are deciding on going out that way. The Pegaseus flights (it's a reliable turkish airline)will go on sale sometime in February for about the same price but with more thinking time.
Transfer time to resort will possibly also a factor for you. An hour will probably be the shortest, 2 hours a comfortable maximum. For Dalaman, this would put into the frame places like Calis Beach and the Olu Deniz area. Really easy to book up accommodation on line either hotel or privately owned accommodation, but do check the reviews thoroughly for the former. Also check distance from the beach/availability of transport to it. October weather can be in low 20's or early 30's so "only 15 minutes walk" in the blurb can turn out to be a family endurance trek! The scenery on the coast from Dalaman is the more specatacular IMO.. i.e. mountainous.
I'm afraid that the last time I visited these areas I was without children and wasn't really clocking the possibilities, but
Flying into Bodrum, the Bodrum Peninsula (anything Bitez and onwards) and more traditional turkish resorts to the east.
One thing to bear in mind is that unless you want it completely quiet, it might be preferrable to visit a resort with a reasonable level of international guests, as the turks regard summer holiday season as strictly late June to early September, with lots closing down after that in the areas where they themselves holiday.
In spite of the fact that it is in a flat valley, with mountains in the distance or the most beautiful of the resorts, I actually bought our place near to Altinkum as it was just so incredibly child friendly for little ones. (lots of child friendly restaurants with playgrounds onsite, safe uncrowde (in Oct) sandy blue flag beaches, with shallow water masses of trips to do within a short radius, including fantastic ancient sites with tunnels my boys love to climb over... that sort of thing. Akbuk, is a lovely small resort nearby (better scenery), much quieter.. I go there a lot for daytrips and would recommend.
On the accommodation issue, I would recommend looking at private apartments/villas. Many owners, like myself have children and you'll find them massively well equipped with DVD players, toys etc. for a comfortable stay. Price-wise, you could get a 3 bed apartment or villa from £150 apt,£250 villa for a week and up. And for these prices you could get sea views if you wanted, close proximity to sandy beaches etc. etc. Certainly a lot cheaper than a package, even allowing for transfers c. £100 return or a car c. £200-£300 p/w.
hope this has helped start you off thinking.