We just returned from a fabulous trip with our 3-year-old and our 2-month-old. After months of agonizing about strollers I decided to leave our double stroller at home and take two lightweight strollers and a baby sling. I'm so glad that I didn't bring the double (we have a Mountain Buggy Urban Double). The MBUD would have been a pain on the plane and on the Disney buses. We also split up several times, especially when I needed to dash over to the baby centre with the baby.
My solution was a Maclaren Quest for our eldest and a Rock Star Baby "Candy" stroller for the baby. The Quest has a partial recline for naps and a high weight limit, which made it perfect for our 3-year-old. The Quest is not suitable from birth, however, and the only Maclaren with a full recline was too expensive to buy just for this trip (we already have a great full-featured single at home). The Rock Star is a reasonably priced lightweight umbrella with a full recline. Just after I bought it Graco came out with an umbrella with a full recline (the IPO or something??) that also looks reasonable. I'll admit that the Rock Star does not push as nicely as a Maclaren, but the baby was so light that it wasn't really a problem.
The umbrella strollers were great for the buses. An umbrella is also compact enough that they let me take the baby's into the table service restaurants (generally strollers have to stay outside, but if you tell them the baby can't sit up and you have included the baby in your seat count when making your ADR they WILL let you remove a chair and park a compact stroller at the table).
I solved the sun issue with a fabulous product called a "Ray Shade". It completely shielded the baby when pulled down all the way. I can't recommend this enough.
I really didn't want the baby to spend the week in a car seat, given that doctors recommend they don't spend more than 2 hours a day in car seats (very hard on their spine). I was very happy with our stroller solution.
Hope this helps!