Since both green teas and black teas are made from the same plant -a plant that contains caffeine- it seems likely both tea variants contain caffeine. The withering & oxidation processes might increase the caffeine in black teas though: http://en.wikipedia.org/wiki/Tea_processing
If you want a caffeine free tea that tastes well (imo), consider Rooibos tea. It contains no caffeine whatsoever: http://en.wikipedia.org/wiki/Rooibos
I like Rooibos tea. Just ordered a huge bag from Amazon. But because it doesn't have caffeine, it doesn't have the same wakefulness effect that tea does.
I also drink coffee sometimes. And even though both contain caffeine there is big difference. When I talk about tea's wake-fullness it is more like alertness, a much more gradual and pleasant effect than coffee. Coffee is like having someone constantly slapping me on the face. It is more noticeable and it is too distracting. But I also happen to like the taste of a good cup of coffee so I drink it then quit after a while, get headaches, go back to my tea and then after a month of two do it again.
This seems like the place to point out the difference between a tea and a tisane: rooibos is a tisane, or "herbal tea," because it is not made from a tea plant.
If you want a caffeine free tea that tastes well (imo), consider Rooibos tea. It contains no caffeine whatsoever: http://en.wikipedia.org/wiki/Rooibos