There are several answers, and the choice is which one do you like to work with best.
Weldbond- can take months to dry clear behind large pieces, no for outside.
Lexel-Very clear silicone based product, but can be gooey-er and stickier than regular silicone and so more difficult to handle.
Silicone-GE 2 or DAP- easy to work with, sets up quickly, a little cloudy when cured.
MAC- expensive, thin and watery, very clear and very strong.

Everyone likes different things. My preference is Loctite silicone. easy to work with, a little cloudy, doesnt bother me. Does some people. A little vinegar smell.

Grout-sanded, other than that, doesnt matter.

The biggest problem with GOG is grout bleed. If the back of the tessera isnt completely coated with adhesive, grout can get behind the glass and show. More likely to happen with textured or uneven glass. Sometimes it happens a little no matter how careful you are. I dont sweat it. However, DAP is a product that can help. the clear version goes on white and then cures to clear. So you can see how well you coated the glass when you apply it. Using a drier mixed grout can help also.

You won't get any better advice than that, even from a professional. Spot on, WACKY.

I use GE Marine silicone(I buy it buy the case!lol) cuz I LOOOOVE GOG!! It dries in less than ten minutes, and is worth the $4.25 a tube that I pay. I call it instant gratification!ha! Speaking of GOG, I was just at Slowmedown's for a visit > OVERTHETOP QUEENDOM!!! and scored a bunch of wonderful mosaic supplies and TONS of Inspiration! so will be posting WIP pics soon!!!
I would add to the advice given, that you want enough glue or silicone to cover the whole back of the glass, but not too much that it builds up around the peice, cuz you need room between the peices to be able to have grout. But it is trial and error, and sometimes I take an exacto blade and cut around the glued pcs so I have room for grout!.

Are you all concerned about the fumes from silicone?

The fumes from silicone are just acetic acid (vinegar) formed as a byproduct of the curing process. Most of the time I barely notice it. Lexel, e 6000 and others have more toxic fumes and I dont use them much except for a bit or a bob here and there. Silicone is relatively harmless. I use a syringe to apply it to the tess and then glue it down, so maybe that is why I barely notice it.

thanks for the great advice... now just to get off my keester and start them... LOL :)

I didn't chime in here because Wacky gave such good advice. I myself use GE silicone 2 clear for outdoor GOG and I'm happy with it. I've also experimented with MAC for outdoor GOG and so far so good.

I came on here to say, "Pick up a piece and glue it on," that's the only way I can get started. I just glue a piece on, even if it's wrong.