For this first method, you only need a coffee mug and a fork. It works if you are new to tea, or if you're in a pinch. Scroll down for more.
IMPORTANT: WATER
Begin with high-quality water. Use fresh, filtered or bottled water that's free of impurities. The water you use can significantly impact the taste of your tea. I personally mix half mineral spring water and half filtered water, using water from a nearby spring.
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OTHER EASY WAY
If you have a coffee press or a nice traditional western teapot you want to enjoy, just double the above - 2 teaspoons to 2 cups of fresh hot water. Share with friends. It's best to pour all the tea out into a pitcher first so the leaves won't continue to brew in the hot water. Good for multiple steepings. [Hack] With practice, you don't have to pour out each infusion, you'll know how much hot water to put back in for just the right tea.
NOTE: Tea balls and similar brewing methods are not recommended.
GONGFU BREWING
Most tea enthusiasts prefer to use traditional Chinese gaiwans (which are simply covered bowls), or "gongfu" teapots. Steeping times are shorter due to the greater ratio of tea to water. Note: Gongfu brewing is mostly for oolongs or darker teas.
Get a nice beginner set up HERE.
A standard-sized gaiwan holds about 130ml or half a cup of liquid. If you'd like to get a good one, look HERE. Warm up your gaiwan with hot water first, pour it out, then add 3-6g of dry leaf, depending on how strong you like your tea. Fill your gaiwan covering the leaves with fresh hot water (195°/90°c), pour out the first dose, then add a second round of water - cover your gaiwan and wait for 30 to 45 seconds. Pour everything into a reservoir through a strainer, let it cool to a comfortable drinking temperature, then enjoy. Repeat this process several times, gradually increasing the steeping time by 5-10 seconds for each infusion. You can find a good reservoir HERE, a good strainer HERE, and some nice cups to share tea with friends HERE.
Traditional teapots serve the same purpose but keep the heat in better. I suggest starting with the same parameters as the gaiwan as a starting point, and then experimenting to see what works best for each tea. The old-school advice is to loosely cover the bottom of your teapot in High Mountain Oolong dry leaf. If you add too much, you'll be busting at the seems. You can find suitable teapots at all price ranges HERE.
As I mentioned earlier, brewing tea is best done with awareness and by engaging all the senses. So, go ahead and enjoy the process!