We took DD when she was 5 months and 2 years old.
When she was 5 months I took her on all the rides in her Baby Bjorn. She was in the front carrier, facing out, and it worked out perfect. I did this on every single ride we went on that did not have a height restriction and those cases we did a baby swap. Having my hands free to get on the rides during loading and unloading (especially on moving rides like Peter Pan, Pooh, and HM) made me feel so much more secure for both of our safety.
When she was 2 she sat on our laps and sometimes sat next to us. She was able to do much more on her own so the circumstances were a bit different than traveling with an infant.