We have four children, and have done Disney both with our own stroller and then with the ones they rent at Disney. By far, it's much easier to rent one of their strollers than to bring your own. They are designed to make it easy to get your kids in and out of them, and the children like riding in them. They should be large enough for your children. If you're concerned, call the Guest information line at ###-###-#### and they will put you in contact with the right people.
May I offer a word of advice? Pace yourself with the kids. Over the years we've learned that doing a few things in the morning that require a lot of energy is a good thing, but only if you can pause during mid-day and have lunch, nap and/or swim in the pool. After taking a break, then going back to the parks is much more enjoyable. There's nothing worse than anticipating having all this fun and then having kids melt down because they're overstimulated and frustrated! If you're staying on the properties, it's good to do the early morning admissions to the parks, and by the time you've done the first hour, the other people are coming in so the lines get longer. It's well worth making sure you start early and then take the mid-day break.
Have a wonderful time! October is great because there are fewer crowds there (well, Columbus Day might be the exception) and the temperatures are mild.