SO SO EASY!!
The first trip we made, 10 days we traveled with an almost 2 year old, almost 3 year old and my youngest was 8 weeks old. We went with friends who had an almost 2 year old, and a 12 week old as well.
The babies were SO easy! We would wake up in the morning, strap them on, and off we went. I was nursing, she would come out of the sling to eat, and have diapers but spent probably 90% of the trip in the sling. Most rides I did not even have to take the sling off, she slept through it all. She was fine at night as well, I felt great it was fine. I have the cutest picture of Mickey holding her, with the other two right there. Mary poppins was going to keep her and let us pick her back up after our epcot day!! We even did a few christmas partys with all of them, it was fabulous. I would take a newborn anyday.
The next trip the youngest was 15 months....while I loved that trip as well, and she was great. It was much easier with a new born, less to worry about.
We did not bring the carseat at all, just a single stroller between all 3 kids, but like I said, the baby spent 90% of the time in the sling, the two year old in the stroller, and my oldest ran everywhere, the entire week. She was like the energizer bunny, just kept going and going!
I will also let you know, the characters (not face), are not supposed to hold babies. But Mickey was knealing down, and asked (in mickey talk), we were right there and it was fine. But, I always tell people not to expect it if they don't, because they are not supposed to hold babies.
Actually on this trip we never even went to the baby care center, no need. She ate wherever we were (on a few slow rides too), and I changed her in the stroller when needed.
Since it is Jan, you won't have to worry about it being too hot either. We went in December, and I was glad we did not have to deal with the heat. Mainly because I sweat like a pig, and I could just imagine wearing a baby in a sling all day in the 100Degree heat. So, it was beautiful when we were there, low 70s, maybe a few days of 80s. Perfect weather.
Have fun!