Like most crafters, I have a plethora of glues in my studio--some are MUST HAVES, and some I rarely or never use. For the most part, I use three types of adhesives: glue sticks, double-sided tape, and white glue. I also always have glue dots and dimensional adhesive on hand.
GLUE STICKS: As long as they are acid free/archival safe, I'll use any brand glue stick. I generally only use them for layouts that will be stored within page protectors, so I'm not too worried about the handling of papers to cause them to pull apart. Glue sticks are the most economical, so using them on large pieces on layouts works well.
ADHESIVE TAPE: Scor-Tape (shown above) is my absolute favorite double-sided tape. It's made in Korea, and I'm not sure how to pronounce the name actually on the roll of tape, but it is distributed here in the U.S. by the same company that sells the Scor-Pad, and they call it Scor-Tape. It works like the common red tacky tape, but has one feature that makes it 100% better in my opinion: Scor-Tape tears by hand easily, so you don't have to grab your scissors ever time you use a piece. It comes in several widths (1/4" is my go-to size), and of course it's acid free/archival safe.
WHITE GLUE: Without reservation, Art Glitter glue is my favorite liquid glue/white glue. It dries in seconds, making it perfect for paper flowers and other 3D projects or for attaching embellishments to cards or layouts. It's clear when it dries, so if you're a little messy, that's okay too. I don't know of a chain store you can count on to carry it, but I'm sure it can be ordered online somewhere. I usually special order it from my Local Scrapbook Store.
One more tip: When storing white glue or other liquids like Close to My Heart's Liquid Glass or Art Glitter Glue, the plastic containers CTMH used to call "Ribbon Rounds" work great. Most glues will fit in them upside down, which keeps the glue toward the top of the bottle. Unfortunately, CTMH doesn't sell ribbon in these cylinders anymore, but if you already have some, this is a great storage solution.