There are pros and cons to taking vs. renting...so you'll get less hassle in one area with one method and in a different area with another. I'll try to summarize my experiences:
Renting:
pros:
1 - big strollers - better for bigger kids
2 - easy to maneuver
3 - don't have to fold up for trams or bus transportation
cons:
1 - EXPENSIVE - $18 per day for a double
2 - no basket for storage
3 - Depending on crowds - sometimes a hassle to get into the park, get the stroller, get the kids situated, etc. I prefer to do this right after getting off the bus so once we're through the gates - we're on our way to the first ride.
Bringing:
pros:
1 - CHEAPER
2 - much better for napping if you have a child who might possibly nap in a stroller
3- Easier to distinguish in a sea of blue rented stroller
4 - Love, Love, Love the basket
5 - Can collapse for rides like WDW railroad
6 - Can take the kids all the way to the bus stop or to the car if they're sleeping (and you parked close enough to walk)
cons:
1 - need to collapse to take on buses, trams
2 - bigger kids might not fit
3 - Maybe worry about someone stealing it (we solved this by getting one cheap from a garage sale - if it gets stolen (very very small chance) - then we rent the rest of the time
You might be able to guess we prefer bringing. BUt, everyone has their own preferences. We're going to check ours through with checked luggage all the way through to our room with Magical Express so it should be easy from that point of view.
Good luck