All green teas have a little caffeine. My favorite is jasmine ("Dragon Pearl Jasmine" is a typical name). I usually use loose teas, meaning you put the marijuana-looking green buds into a tea ball (a metal mesh tea ball) and drop it in. You can also use a coffee filter over the top of the mug and pull it out when it's done, if you don't want to invest in a mesh tea ball.
The key to tasty green tea is very hot, but NOT boiling water. If you use boiling water, like what you use for black teas, the tea tends up tasting very bitter and grassy. If you use hot water, it's perfectly fruity and light.
And you can always add a drop or so of honey or sugar to sweeten it up if it's still not to your taste!