First of all, are you planning to free-range in your backyard? If so, you can have more birds, but be warned, they'll make a mess on your porch!
If not, you'll need to build a run in addition to your coop. The coop regardless should have about 4 square feet of space per bird. If you live in a really cold area, I've heard a little smaller coop helps them keep warmer; I wouldn't know, living in Arizona. The run should be 8-10 square feet per bird. Ours is a bit smaller but we let ours out to free-range frequently. Ultimately, I'd make it a bit bigger than you think you need. Chickens are like potato chips, you always end up having more than you planned on.
Good choice on the breeds. I'd also suggest Buff Orpingtons, they're just my personal favorite, but our flock is mostly those and the Barred Rocks & Rhode Island Reds you mentioned. Great layers, pretty healthy. Barred Rocks tend to have more personality than Rhode Island Reds, though maybe we just ended up with birds like that.
If you're free-ranging, you easily have enough space for a starter flock of 10 birds or so (I'd start with no more than a dozen if I were you, you can always get more in a few months). Whether you free-range or not depends mainly on local predators and, as I mentioned, your personal objections to mess. Ours make quite a lot of mess, but we have 30.
As for a rooster -if you have 10 or 12 birds as suggested, you'll only need one. The general rule is at least 8-10 hens per rooster. For years, our mutt managed a flock of 20+ by himself. The one Barred Rock roo we had was so mean we had to rehome him.

That doesn't mean yours will be, though. Buff Orpington roosters are supposed to be nice (sorry to keep ragging on that breed, they're just the best IMO) but I've never had a rooster that breed either. All our roosters have been mutts we've taken in...roosters are easily found online for cheap. It might be easiest to get an adult that you already know isn't going to grow up mean, or grow so big he'll hurt your girls. Other than that, the breed really doesn't matter.
If you start out with chicks, I'd get an adult roo once they're already full-grown. For that matter, do you need a rooster? If you're not planning to breed them, it's not necessary to have one.
Sorry for the essay! Hope I didn't make it sound intimidating. Honestly, the hardest part of chickens is the stage you're in -the planning. They're fairly sturdy, happy animals and you'll have a lot of fun with them. Good luck!