Instant adhesives are generally in the family of cyanoacrylate adhesives -- although there are two basic types: those that cure by very quick evaporation of solvents, like Crazy Glue found in local hardware stores -- and those that cure by depriving the glue of oxygen, like PascoFix. The advantage of oxygen starvation for creating a bond is that the liquid will not evaporate quickly while applying it, but as soon as you squeeze two well fitting non-absorbing surfaces together, the bond is instant.
Quite frankly I have had little success with the standard super glues -- I have had great success with the more difficult to find PascoFix. You can even buy powder "fillers" that together with the liquid allows for patching holes as well as filling gaps.
To remove spilt glue, or to un-glue things (like fingers) -- the secret is acetone. Finger nail polish remover used to all be acetone based and although there are "acetone free" removers now available if it doesn't say that, it is still acetone. You can also purchase bottles of acetone from paint stores. First check that the acetone itself will not melt or stain the surface you want to work on. If a drop of acetone in a discrete corner of the counter top or other surface you want to work on has no effect, then simply flood the glue spill with acetone and wipe it off.
To un-glue things you need to work the acetone into the glue joint -- not always easy with two rigid objects and they may require some soaking. For flexible materials (like stuck together fingers) gently work the acetone into the edge of the joint as the two parts slowly separate.