Which type of adhesive should I use on posters, and when?
Assuming you're trying to affix things to poster board paper, here are some things:

Glue sticks


They don't set super fast, which is a pro and con:


You can adjust your placement if you make a mistake, and have a fair amount of time to do this
You have to wait until it's 100% set, being careful not to push or slide it around 

They don't have the best sticking power, and items may fall off over time
You can apply it to either the board or the object you're placing very easily
It works best on flat, non-porous surfaces
It can be removed very easily if you accidentally put it somewhere, with a quick wipe of a dry or slighty-moist towel
It comes in colored varieties that dry clear, now, which helps you make sure you've applied it properly
It doesn't wrinkle thin paper as much as liquid glue, if at all


Use glue sticks if you want to put things up quick, have leeway with mistakes, and need a steadfastness in the sticking power

Liquid glues

Assuming you're applying it out of the bottle, and not using a brush


They take a bit of time to set, but have much better sticking power than glue sticks once dried
You can't easily adjust a misplaced item without smearing the glue around
They are more easily applied to the board, on a flat surface
They can work on porous materials, such as felt, and odd shapes such as pasta


If too much is used, they will bleed through the felt and make it stiff on the opposite side 

It's a bit harder to clean up if you put a drop in the wrong spot, as it wiping will spread it
They dry clear, generally
If you're applying thin materials, like tissue paper, magazine pages or printer paper, then it's very likely to cause some wrinkles


Use liquid glues when you want your project to stay together longer, have some time and space to wait for it to dry, or want to use a variety of materials.

Tape

I'm assuming Scotch tape/cello tape, for this.


It sets instantly! Apply it and you're done.
If you make a mistake placing your object, you have to risk tearing the paper to pull it off ad try again
The sticking power usually depends on the length of tape you apply
You can fold a strip onto itself to create a double-sided tape, so the strips don't show on the outside


This takes more time and material and makes the tape easier to remove (a good and bad thing, depending)

The object you're sticking needs to be flat and thin, if applying it from the outside, otherwise it may pull of or not stick at all
It works best on clean edges, and not the kind made with decorate edge scissors, such as zig zags
You can find a variety of similar products, such as washi tape, to spruce up your project
It doesn't wrinkle your papers, but you have to avoid getting wrinkles in it


Use plain tape when you want something up fast and are willing to risk some tearing (that might end up hidden) on parts of your project and don't mind the obvious appearance of tap when using it on the outside. On the other hand, if you want to add quick, decorative borders, you can use washi tape or other products.

Double-sided tape is available, if you're using it under the object your applying, but it doesn't necessarily have extra sticking power than rolled up regular tape. Yet, it tends to be more expensive. 

Another alternative to tape is stickers.