We went when our second child was 2 months old. We also had a barely 3 year old with us though. A 9 week old is easier than a mobile child in many cases. I didn't use the baby care centers at most places but just nursed wherever. I think the only park I used one at was Sea World. So, no comments on how the baby care centers are, but if you choose to nurse in public, it's not a big deal there at all.
Get some sort of carrier if you don't have one already. I used a ring sling, like a Maya Wrap, and my daughter was in it pretty much all day. The barely 3 year old used our stroller. She was happier in the sling and it made waiting in lines easy. Do take a stroller anyway, as some babies will nap better in the stroller - she did at 8 months but not at 2 months. Take one with full recline. I used the sling opened up over the stroller to block out the sights to help her sleep too. Again, that was at 8 months and not needed at 2 months for her - but I can't remember if 2 month olds need that sometimes. If she's not used to a carrier, then try nursing her in it to get her used to it ASAP.