to enter a slime biome:
build a couple storage units under water. put 9 priority on them, with only slime/polluted dirt.
place 1 or two deodorizer's at dig site.
plan dig pattern to trap polluted oxygen at outside of base for a bit. add Deodorizers in polluted oxygen.
Build a washstation between dig site and base, set to dupes wash themselves on returning to base.
Take your time digging. Allow deodorizers to work and clear out polluted oxygen and germs in air. dont make dupes run back into the germs over and over.
To enter caustic biome.
build an airlock, so hydro rises, and chlorine sinks, and neither enters your base.
Build insulation tiles, as this biome is likely much warmer then your base, and you probably don't want it to bleed in.