What glue should I use to make my first fairy garden?

Elmer's glue has an App - where you put in the items you are attaching & it will give you recommendations!

Use Loctite which is a epoxy glue for outdoors and dries clear.

I use EverBuild StixAll for all our fairy gardens. I'm not sure if it's available in the States though. It's tacky so the the stones will stay put which is a big bonus when you're trying to glue them onto round surfaces. It's also super strong, dries clear and it's water resistant. Been using it for years and I've never been disappointed.

the sealant I find to be the best is Lexel, it's clear, water proof, mildew resistant, and paintable. Sticks to most surfaces. I've used it on glass, plastic, metal and more

You could use clear caulk, E6000, look for any waterproof type glue unless you want to go with something like quickcrete.

http://elmers.com/glueguide