Maybe you're overbrewing it. Green tea only takes about 2-3 minutes to brew in water that is about 180 degrees (ideally). for the right brewing temp, you probably want to fill your cup up with about 3/4 boiling water (fresh out of the kettle), and 1/4 room temperature water (out of the faucet or whatever). while brewing, don't disturb it, just let the tea bag sit, don't fuck with it. Don't move it up and down. Don't swing it in circles. Go read a book or check your email or something. and NEVER milk your teabag when your done. Squeezing the bag when you're done brewing it is going to get all of that bitter nastiness into your drink.
Free teas are often better than bagged teas, and to get the most out of them you could use a french press or teapot with a strainer built in.
Lastly, there are different green teas (a lot of them). Dragon tears, gunpowder, gen mai cha (my favorite), the powdery extract type stuff, etc. It may take a while to find your favorite. if you've gone through countless teas and can't find one you like but still want to consume it for health benefits, you may want to try taking the powdered extract and putting it in capsules.
