When you are in St. Louis, you should ride part of the Katy Trail. It starts in St. Charles and heads west across the whole state, following a railbed near the Missouri River. The surface is crushed limestone. It would be easier to rent a bike there (lots of rental places in St. Charles) than bring your bike with you. To take your bike with you, you'd either have to invest in an expensive bike crate, or stop at a bike shop and pick up a used cardboard bike box. The bike has to be partly disassembled to pack. This involves removing the front wheel, saddle and seatpost, stem and handlebars, and pedals. It takes some tools and technical savvy to accomplish this yourself, or you could pay a bike shop to pack it and unpack it. Then you have to worry about the airline damaging your precious. I'd suggest renting if you only want to ride a few times. If you really plan to ride every day then it might be worth taking your own bike.