Boots work for most of your weight workouts. Exceptions include deadlifts (barefoot/socks or in deadlift slipper is ideal). Squats (boot with a slightly elevated heel). Any calf exercise - you wanna trainer for these.
Cardio, of any kind, requires a proper training shoe. Brands come down to preference. A proper sports shoe place will measure and fit you on request.
A few buddies have tried the Virbram 'feet' style shoe.