I recommend Stash Tea—they have sampler packs of loose tea, and their tea is quite good. I echo Gail’s recommendation to avoid tea bags—the tea used for teabags is often of lesser quality. And you can control the strength of loose tea. It does help to have equipment—either disposable paper teabags, or infusers, or a proper tea pot.
If you’re just starting out with tea, or you’re checking out a new tea company, go for the basics—Irish Breakfast and English Breakfast are blended black teas, and you can get unblended Ceylon, Darjeeling, and Assam teas from most reputable tea shops. If the shop can’t make a good Irish Breakfast, they’re unlikely to get anything more complicated right.
You might also like flavored black teas – Earl Grey is the most well known one, but just about any fruit has been dried and added to black tea, and some of them are quite tasty. Some people look down on them, true, but they can brew their own tea if they don’t like yours. And Earl Grey is another one of those basic teas where if the shop can’t get it right then you should probably avoid their other teas.
(Generic teas are just “Darjeeling.” Higher-end teas will tell you what area the tea comes from—“Darjeeling Okayti.” The difference between black/oolong/green/white tea is how much oxidation (sometimes inaccurately called “fermentation”) has gone on—black tea is left to sit for a while before drying, and so the chlorophyll breaks down, while white tea is dried almost immediately.)
I’m also fond of jasmine tea—it’s green tea to which dried jasmine flowers have been added. You can get very frou-frou varieties of jasmine tea where the flowers have been carefully dried so that they unfurl in the teapot to make pretty decorations, but that’s overkill.
And there are “herbal teas” or tisanes—in some idiolects, including mine, it’s not tea if it isn’t some variety of Camellia sinensis. These are basically dried fruits, herbs, and other flavorings, and because there’s such a variety it’s tough to generalize. It’s worth distinguishing between tea with added flavorings (such as Moroccan Mint, which is usually green tea and some blend of peppermint and spearmint) and tisanes of mint, which have no actual tea in them, just mint leaves.