Do you wear underwear with padded cycling shorts?
No. Cycling shorts are designed to be worn directly against your skin.
Underwear creates extra seams, layers and folds that will move separately from the shorts. Over time, those small movements become friction, and hot spots are exactly what the chamois is designed to reduce.
It's one of the few occasions in life when "leave the underwear at home" is genuinely good advice. You'll probably forget you're not wearing any after ten minutes because everything simply works better without another layer getting in the way.
Bib shorts, waist shorts or padded liners?
All padded cycling shorts do the same basic job, but they're designed for different styles of riding.
Bib shorts use shoulder straps instead of an elastic waistband. The straps keep the shorts and pad in the same position throughout the ride, so there's less chance of the pad shifting as you move around on the bike. They also remove the pressure of a waistband when you're bent over the handlebars.
Waist shorts are simpler and usually more affordable. They're a good choice for shorter rides, commuting or anyone who prefers not to wear bib straps. If they fit properly, they can still keep you comfortable for several hours.