For the most part, I use glue sticks...I really like the prang purple large glue sticks. This is great for beginning kinders as the mess is so much less!!! Then I move to glue in little cups with lids (I have a friend who brings me these great 1/2 cup plastic containers w/lids from her chinese food!!!) I fill them about 1/2 way and use little brushes to apply. I use white glue in the cups...and as the water evaporates, the glue becomes like tacky glue. I use this especially for collages that use fabric, found objects, or heavy papers.
I also bought a set of low temp. glue guns....I use these occassionally with 4th and 5th graders for very specific kinds of projects.
Other kinds of adhesives I like are: Quick Grab....like a gel epoxy...it is great (toxic for little ones)!!! and....Epoxy Putty...you can buy it at Home Depot in the plumbing section...and it is wonderful for putting hooks on the back of ceramic pieces, etc.....dries like cement...but toxic for little ones to use!
For some things...I also like doubled sided tape...but only Scotch Removable Poster Tape...expensive..but worth it if it is what you need! And last but not least...for hanging work on cement/tile walls....DAP (gotta be DAP no other brand works!) Fun Tack!!! Use in huge globs!!!