I agree with the previous posts. Most of my experience was with hatchery based Rhode Island Reds, or production Reds as you will hear them called. You seem to be looking for exactly what I was last year. A year later, here are my observations.
To me, at least, having chickens for meat and eggs implies some sort of self sufficiency in your family, so why not look at your flock the same way? You will quickly find that virtually all hatchery based poultry lacks broodiness. If you are new to chickens, broodiness is like mammals going into heat. Its the female desire to raise young. Sure they will lay eggs, but will they set on them and hatch them out? Playing with an incubator can be fun, and informative for small children. No matter, a momma hen will raise chicks better than any of us, and will leave our time free to take care of other tasks.
Some others will add to this short list, but game fowl, Cochins, and Silkies are about the only backyard poultry that will go broody. Cochins are probably the only of them with any real meat qualities, and I don't know about their laying. Many people will use them and silkies as surrogate mothers to raise mixed flocks.
There is one other oddball possibility. There is a large, meaty chicken with better than average growth rates, that lays 3-5 eggs a week, and depending on the strain will go broody. Its the Marans, originally bred in France as a dual purpose chicken. Later they became well know amongst chefs for the meat qualities and for those wonderful, dark chocolate colored eggs. Yes, you may here from many that they are just a fad. Right now, we still have enough stock that hasn't been too perfected so that growth rates are good, and hens will occasionally go broody. Another advantage is there temperament. These calm and docile flock managers also make great crosses with other breeds for meat purposes. Personally, I can vouch for a black tail buff Marans crossed with production reds. They do actually grow faster, and make beautiful offspring.
If you choose to talk to breeders, make sure you ask about growth rates and broodiness. These are the hallmarks of great dual purpose, back yard chickens that seem to be the first things lost in modern animal husbandry.
Choose wisely. Only you can make the best choices for you and yours....
Ken