i've been there twice now, had a great tailgating experience each time. since the nfl won't let you into the lots until 3 hours before kickoff, chief fans literally start their tailgating while driving into the stadium. no kidding, we saw numerous grills and smokers being fired up in the back of open bed pickups while idling in the lines waiting for the gates to open.
i lived in KC for a year - would not really recommend downtown as this is a city that defines midwest urban sprawl... there is frankly not a lot of action right downtown unless you like overpriced shopping at crown center or on the plaza (which is actually several miles from the big buildings)
gates barbecue, however, is well worth the trip to main street in downtown (best working class bbq in the city). also the negro league baseball HOF and the jazz museum at 18th and vine are good to visit (one ticket gets you into both) if you get into town early and have time to kill on saturday