I say go for it...I find that babies are very portable...mine sure has been! Babies aren't that fragile...unless your child is sick, I don't see any reason why the trip wouldn't be doable. DS flew with us at about 2.5 months and was a complete angel...even with a connection each way, he didn't fuss a bit. Heck, most of the time he was asleep by the time the plane took off. Gotta love the "white noise" effect of jet engines.
1. Check your car seat owner's manual...you might not need the base. Most can be installed with or without the base. You might also want to check Consumer Reports for their safety ratings for your particular seat with and without the base. (they give ratings for both uses on seats they test)
2. Taking the stroller for the flight would be easy. You can gate check it at each flight. Just be prepared to take baby out of stroller and infant carrier at security...you'll have to put the stroller and carrier through the x-ray machine. (This is true almost all the time. Once when I was traveling with DS and no one else, they let me wheel the stroller through and they hand-checked it...but that was one time in 10...yes, my 1-yo has flown 10 times!)
Gate check works like this. At the gate, ask for a gate check tag. Sometimes they give them out before boarding; other times they give them out as you board. If you are on a small flight where you have to walk on the tarmac to the plane, you will likely get the tag at planeside and a gate attendant will take it. It will be waiting for you planeside when you land. If you are boarding via a jetway, you leave your stroller (folded with gate check tag on it) at the end of the jetway by the door. It will be waiting for you when you deplane...sometimes they even unfold it for you.
3. What travel system do you have? If you can manage, you might consider getting a Combi convenience series stroller. This is a very lightweight stroller that folds compactly yet has a bar that accepts all major brand infant carriers. They aren't very expensive, and much easier to deal with IMO than the average Graco/Evenflo/Century travel system stroller. You can find great deals on Combis at
www.strollers4less.com or
www.combistrollerstore.com. Look for models that have the "Perfect Match" attachment bar...these are the ones that work like travel system strollers.
If you go with the standard travel stroller AND you have a lot of luggage (as most people traveling with a small baby do!), be sure you rent a car that will hold all your luggage and your stroller.
4. Can you use a front carrier like the Baby Bjorn? If so, you might find this much preferable in the parks than a stroller at times. You have your hands free and baby can stay close to you and probably warmer. We loved our BB and were sad when DS eventually outgrew it.
I hope this helps! Good luck and happy travels!