Wow, the Katy Trail (a 264-mile trail that winds through Missouri, passing towns, wineries, and B&Bs) looks fantastic! It has plenty of distance for sure, but its an excellent trail with gentle grades. Overall, this is an excellent objective.
So, what bike? A road bike would be too uncomfortable. A mountain bike is too heavy and has too much gear you dont need (rear suspension, for instance). So definitely look at some kind of hybrid/city bike.
One intriguing possibility is the Marin San Rafael ($510). Its a city bike with a few mountain bike touches, such as a light front suspension fork that would soak up the bumps you might encounter on some portions of the Katy. It also has a suspended seatpost for a cushier ride. The frame is light aluminum, and it has plenty of gears for steeper terrain. For a comfortable tour of the Katy, the San Rafael would be terrific.
I also like Giants Tran Send EX ($760). It, too, has a light front suspension fork, good gearing, and a light aluminum frame. And it comes with a full travel setupfenders, rear rack, the whole deal. It would be a great trail bike and a good fitness bike for you moving forward.
For a bit more speed but still good comfort, take a look at the Cannondale Road Warrior 2 ($900). Its a bit more of a mix between a road bike and a city bike, with upright seating and bars but skinner tires, a little more forward lean," and better components. And its designed to handle all sorts of road/trail conditions, so it would negotiate the Katy in style.
As for gear, youd need the usual that you'd buy whenever buying a new bike: helmet (Bell Furio, $64), rear seat bag (Cannondale SeedPod, $35), rear luggage rack (Jandd Standard Rack, $48), tire pump (Blackburn CF, $44), and some panniers for clothing and the like (TransIt waterproof panniers from Performance, $70). Plus tubes, patch kits, that sort of thing. What kind of bike clothing you get is entirely up to you. These days there are lots of comfortable options that dont swaddle you in Spandex, such as the Zoic Posh Short for $45. Otherwise everyday outdoors wear is fine.
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