I know halogen lamps have iodine or bromine in them and I know regular incandescent lamps are filled with nitrogen.

But what exactly are krypton and xenon bulbs? What advantage do these gasses have over nitrogen? Does it have something to do with them being better insulators? Can halogen lamps have these two gas fills?

Yes, it has to do with their insulating properties. Krypton and xenon (especially xenon) have lower heat conductivity than nitrogen, which leads to lower heat loss via the gas fill and thereby greater efficiency. Also, the atoms are heavier, and better at protecting the tungsten atoms in the filament from evaporating, by bouncing them back at the filament. This enables the lamp to be driven at a higher temperature (and therefore higher effiency since more radiation will be in the visible region) or longer life for the same efficiency.

Standard incandescent lamps do not have pure nitrogen, but some argon mixed in to reduce the risk of arcing when the filament fails, at least in 220-240 V lamps.

For halogen lamps: they always have a gas fill like standard incandescent (which may be krypton or xenon), with the addition of small amounts of a halogen (bromine, iodine).