The Russian tea ceremony, while not as ritualized as the Japanese tea ceremony, is an extremely imporant part of daily life in Russia. There are two steps to drinking chai like a Russian: making it, and drinking it.
Making tea like a Russian. You will need to get yourself:
a smallish teapot (or chainik, in Russian) - preferrably one with a filtered spout, but this is not necessary.
loose leaf tea is a must, but does not have to be actually grown in Russia.
Now that you have your supplies together, you're ready to brew the tea. Fill a teakettle with water and set it on the stove to boil. Let it sit at full boil for at least 5 minutes while you sit in the kitchen talking, completely oblivious to it. You can even let most of the water boil off, refill it and repeat a couple of times if you like - but please be careful not to leave it long enough to start a fire and burn your house down.
Once you notice that the teakettle is ready, turn the heat off and pour a little bit of water into the bottom of your teapot and put the lid on. Let it sit for a minute or two to get the teapot warm, then pour it down the sink. Now spoon a bunch of tea leaves into your teapot (you should be aiming to have a teapot full of double strong tea) and fill about halfway with hot water. Put the lid on and let it steep you some very black tea. After about 5 minutes you can top the teapot off, and your tea is ready.
Drinking tea like a Russian: You will need to have some sort of sweet rolls, bread, cake, or cookies (in the English biscuit sense) on the table to munch on. If you haven't got anything in the house, send someone out for it. If anyone has said that they're already full, make sure you target them with your attempts to get people to eat the sweets. Also note that to be truly Russian, you should leave the spoon in the mug while you drink, cake should be eaten with the teaspoons or by hand, and that dipping an already tea-wet spoon into the sugar bowl is perfectly acceptable.
When pouring the tea remember that its brewed double strong, so ask how strong your recipient wants their tea. A half cup topped off from the teakettle should yield a 'normal' cup of tea, but you can of course adjust how strong your tea is by adjusting your ratios.
Basic tea: Black tea with sugar. You should add three heaping spoonfuls of sugar no matter what size the cup and drink the tea while it is still scalding hot.
Tea with Lemon: Cut yourself a thin slice (not wedge!) of lemon and put it in the bottom of your cup before you pour your tea. If you want more lemon, don't cut a thicker slice, instead add more of the very thin slices, and again, lots of sugar. This is how tea is drunk on trains.
Tea with jam: Put some jam (I recommend black currant jam if you can find it, however the Soviet classic is bad strawberry) in a tiny little bowl and eat it by the spoonful (the tiny little teaspoon-fuls) in between gulps of your scalding hot tea. It is acceptable to leave your tea unsweetened in this scenario.
If any milk product comes near your tea, you are no longer drinking russkii chai, you are now drinking your chai po-angliski.
Note that if you have children in your party, they should also drink tea. Make it very weak, very sweet, and pour it into a tea saucer for them to drink from, so that they don't scald themselves.
After tea, make sure you leave the leaves in the pot. next time you make tea, splash a little extra water into the teapot before you warm the teapot up, and dump the leaves into the toilet. Do not flush.
Repeat the entire process at least twice a day, and whenever you have guests.