The kit lens will do well for you for the majority of your photo needs. If you want to get more into the photography aspect instead of just taking snapshots, then the other lens or lenses that you'll need/want will depend on your shooting style.
Some people can go with just one lens and be happy. Many will take 2 or 3 lenses such as the basic kit lens like you have along with a telephoto lens (50-200 or 70-300) and also a very fast lens like the 50mm f/1.8 or 30mm f/1.4 for the indoor rides.
A small group (such as the regulars here on this photo board) will go with as many as 5 or 6 lenses. In addition to the lenses that have been mentioned they may take an Ultra Wide Angle like a 10-20mm or 11-18mm (or something similar to that range) or maybe a Macro lens.
I typically bring with me 4 lenses and use them for the type of picture I'm looking for at a given park. I mostly use my telephoto lens (a 70-300mm) at Animal Kingdom. I'll use my 30mm f/1.4 at MK and DHS for the indoor rides (occasionally at EPCOT too). I like the UWA lens (I use a Sigma 10-20mm) at all the parks for a different perspective and my standard walk-around lens goes with me to all the parks (I've used an 18-70mm, 18-135mm and am currently using a 17-50mm f/2.8, but would also use a 24-70mm f/2.8).
So basically, it depends on the shooter.