Tea in Chinatown
I haven't been there in awhile but I think kam man on canal between mulberry and mott sell lots of boxed teas as well as teas from large glass jars in the basement. Not sure how user friendly it is but I remember last time they allow you to open the glass the jars up and sniff the tea or something.
For Taiwanese tea (high mountain oolong) there's an additional Ten Ren on Lafayette between Howard and Canal.
For Mainland Chinese tea (Te Kwang Yin types as well as Chinese green teas)On Howard street(between Lafayette and Broadway), just a few steps walk from the new Ten Ren is the is the Big Apple Tea House that started as a Chinese tea cafe, with special tables and teapot/tray/boiler/strainer...etc for the customers to make tea at their own table. These days they have changed somewhat to include other stuff such as gift ware. You might check to see what they have before they completely go under.