Ted, first of all a Green Mango is technically any unripe mango one finds, but particularly there are a couple of cultivars that are intended specifically for unripe harvest and consumption - they have very little fibre (stringy stuff), a higher volatile oil content, and fairly small pits, and are sort of paisly shaped. Personally, I can't stand the true green mangoes when they're ripe - they're so strong by then that they can be used to strip paint.
This said, I peel the mango and chop it into cubes or julienne it, then soak the pieces in lemon water until they're ready to use (this keeps them nice and pale and keeps them from going limp). They go into the stirfry like any other veggie, generally at the same time as hard veg like celery. I normally add a bit of powdered chili pepper to anything containing green mango, because the two flavours complement each other nicely. If you have one of those rotary apple-peeler machine thingies, you can also sort of spaghetti green mango and use it raw with a bit of salt instead of pasta.
You can do the same thing with half-ripe mangoes, but the flavour is very different since the sugars have developed more fully.
Green mango (at least, the green mango I'm familiar with) has a tart, agreeable flavour similar to a granny smith apple with hints of pine. It's also a popular snack here simply cut into long slivers and dipped in a mixture of salt and chili powder - there are mango carts all over the place right now, because it's mango season.
Part of your problem might also be that you're stuck with the bog-standard Tommy Atkins mangoes, which are hideous even when they're ripe (again, I'm spoiled for choice here, but they really are skanky). Try cruising your local Oriental and West Indian produce stores - they'll have a host of different types available. In particular, keep an eye out for the gold-skinned/red blushed 'Julie' and 'Juliet' cultivars (very flavourful even when a little green), and the all-gold, tiny 'Ambassador' (which is a fibreless mango used as a natural juicebox - you mash them up between your hands, then bite a little hole in the bottom and just slurp the juice out.) 'Tis the season, so with a little luck you'll find something tasty.