See if there are any brick and mortar shops in your area (unless it is Teavana, they are cancer), if the salesman is worth his salt he will make everything much easier and probably quite fun. If not you are best off with a good online shop; I've heard good things about Verdant Tea.
Tea can best be divided in to a few categories- White, Green, Oolong, Black (or red as it is called in China), and Pu'ehr, plus of course various herbal infusions.
I'd recommend starting with a set of a Chinese or Japanese green, a Chinese Oolong, and a Chinese black. White is typically a bit more expensive and less versatile due to its mild flavor. Pu'er is also expensive, but it also tastes unlike anything else so I'd hold off on acquiring any just yet.
Japanese greens are typically grassy and vegetal in flavor profile while Chinese greens are more floral and complex. There is still a great variety between the two overarching types though. While I enjoy jasmine and osmanthus teas a lot, you are probably best off with an unscented variety so you can enjoy the actual tea rather than the flower, to start at least.
Oolong is slightly to moderately oxidized, with no narrow parameters, so there is a huge variety of flavors to explore, ranging from floral to woody to fruity. My personal favorite is Quangzhou milk oolong, which tastes uncannily like of all things warm vanilla milk.
Blacks are fully oxidized and thus have a very different flavor from the other varieties. Blacks are typically very robust with flavors being very complex, typically woody and smokey hints with a distinct sweetness. Western blacks are usually blends of oriental varieties in order to provide a consistent flavor after many seasons, often optimized for drinking with milk and sugar- so I'd advise against going down that route to start.
White teas are minimally processed and as such have very delicate and mild floral flavors. They are usually a bit harder to find and a bit more expensive. The light flavors make it unsuitable for pairing with food in most cases, or may simply be too weak to be enjoyable for some.
Pu'ehr is completely different from all other varieties, as it is fermented to varying degrees. it is also rather expensive and the flavors are very peculiar, unlike most anything else.
With brewing you are best off doing it Gong-fu style, which is with a large quantity of leaves, a short brewing time, and a small amount of hot water. This allows for many infusions, so don't worry about using up too much tea. You can get up to ten good infusions from one set of leaves. Western brewing isn't inherently bad, but it is easier to screw up.