You rent strollers inside the turnstiles, through the right hand tunnel and to your right. Yes, it's usually mobbed first thing in the morning, and you can waste time standing in line for this. However, you can pay for multiple days on your first day. Then on subsequent days, you bypass the rental line, and just walk up to the CMs handing out the strollers, and show them your multi-day receipt, and they'll hand you a stroller. Keep your stroller nametag (fits inside a clear plastic window on top of the stroller canopy for identification) that they made out for you on the first day and it will be even quicker.
You can take them on the Esplanade, and between both DL and CA. However, you can't take them out past the security checkpoints, so that means if you're walking back to your hotel or to DTD, you're going to have to carry the kiddos or have them walk.
If you leave the park during the afternoon for a break, then come back later, you can just show the receipt again for a new stroller. However, we just grabbed one from the tons of abandoned strollers right next to the security stations.
We liked the rental strollers, they were sturdy and roomy (our almost-6-year old was quite comfortable and it saved her legs!) and they steered very well.
Hope that helps!