There are tea growing regions in India (and China) (to name the two most famous producers) (but all over Asia and also in Kenya) just like there are wine regions in France. Darjeeling and Assam are deservedly the most famous in India. The regular tea in bags is the small dustings of leaves. You can also get loose leaf tea in bags. The loose leaf is much more flavorful. There are two major tea seasons: spring and summer (called first flush and second flush, respectively). There are also autumn and a few winter teas. The same bushes from the same estate will produce different teas based on time of year. The teas from different estates have different characteristics. There are four major types of tea: white, green, black (my favorite), and oolong. China also makes a puerh (fermented and aged) tea. Japan is famous for a ground green tea called Matcha.
There are high-fired teas that are smokey. Then there are herbal "teas" (also called tisane) made from all sorts of herbs but not tea (so they are non caffeine). There are blended teas (mixed from various estates) and flavored teas, where different herbs, flowers, or spices are added to the tea leaves, the most popular being earl gray (bergamot, a citrus plant) and masala chai (any number of Indian spices: cinnamon, cardamon, ginger, etc.). I buy tea straight from India. I stumbled into that a while back, starting with bagged tea, trading up, reading up, and discovering that my favorite supermarket tea, Irish breakfast, used predominately leafs from Assam, India. So I just started getting it from there (actually from a distributor in Darjeeling). They had a plain little website, which has been redone a couple of times and now is fairly posh looking. But cities will usually have tea shops or coffee shops with a decent collection of teas. There are all sorts of blenders who mix and label their own brands. (The one SmokeyTom posted looks good.) I don't really know anything about the Chinese teas except that Yunnan is the nearest relative to the Assam from India that is my favorite. It's just like any food thing of the sort, beers or wines or whiskeys, cheeses, etc., there are all sorts of variety and qualities and it's fun to play with. I bet Nick has the skinny on places to order in Asia. But I would start with good, easy American and English brands and blends to learn your way around first.