I'm a native Kansan actually moving back to KC for a year for my husband's work, and I've lived in lots of different parts of the country, so I think I have a pretty good perspective on the pros and cons of KC. Frankly, I think it's an undiscovered gem, and I hope it stays that way!
Weather-wise, I think it's better than some of the places I've lived (S. Georgia, similar to Fla), because I find hot and humid to be more uncomfortable than cold. It's true though, that it can be really frickin cold in the winter. And it is hot in the summer- but this peak of heat is much shorter than Fla (and central Tx, another area I've lived).
Culturally, it's a fabulous city! Lots of highbrow and lowbrow options, and if you're willing to step outside of the box, fairly diverse as well. It's a city with a lot of history, which I still enjoy learning about, even though I grew up there. Overland Park has a history of being really "white," but I think that's changed some, and other parts of KC are really culturally diverse and interesting (the historic jazz district has been built up over the last few years, and is amazing). Housing prices are comparatively low, and the schools are terrific, despite the state school board's attempts to the contrary in the last few years (sorry, just a little politicizing).
In terms of recreation, yes there are definitely areas for hiking and camping, and good access to some pretty lakes. But you'll have to look a little bit, if you're used to North Georgia-type camping- no mountains here). The Ozark Mountains are a relatively short trip, and they're really beautiful.
I hope you enjoy it! I too have fond memories of growing up in NE Kansas, and agree that the bbq is amazing. Very different than bbq in the southeast though, and no Low Country Boil.