Drinking whiskey is a lot like getting into a romantic relationship. Your question leads me to believe you're looking for a mature, long lasting relationship with your whiskey, not just a quick shot. Do remember that you're not in a hurry to get drunk, you want to taste everything this drink has to offer. Take your time and get to know your drink.
- You both need room to breathe. Use an old fashioned glass, which is an actual type of tumbler, not an antique. The glass is as wide as it is tall, and give the whiskey lots of air. It also lets you get your nose in the glass so you can smell all the amazing stuff happening in there.
Keep things one-on-one. If you really want to learn to detect different flavors in your whiskey, don't mix it with sodas like Coke or Sprite because the sugar and other flavors in the soda will overwhelm the flavor of the whiskey. (It's fine if you like drinking it that way, but it won't help you do what you asked.) Friends are nice in small doses. Water is whiskey's best friend... in small amounts. If you've got cask strength whiskey in a glass, it's going to burn you. Splash a little water in there and you'll be amazed at how much flavor opens up. If you're just starting out, try mixing 3/4 whiskey to 1/4 water in your glass. This will take the edge off the alcohol burn and allow you to taste the other flavors. Don't overdo it to start. You can always add more water if the drink's too strong. If you want more bite, finish what you've got and add less water when you pour the next drink. Carbonated water is a whole other experience because the bubbles change how the flavors reach you, enhancing some and muting others. (If you're used to drinking your booze with Coke, try it with fizzy water sometime.) Break the ice. Ice can be good if you're just starting out and find whiskey a little overwhelming, because the cold cuts back some of the flavors and the ice dilutes the drink as it melts. For seasoned drinkers, ice is a very personal preference, but I avoid it if I want to really taste a whiskey because warmth releases a lot of the flavors. Dance with your partner. Swirl your whiskey in the glass for a minute. This will allow some of the alcohol to evaporate and carry the wonderful smells of oak and vanilla to your nose. If you smell only paint thinner, swirl a little more and smell again. Keep doing this for a little while before you take a sip. EDIT: Adey Hill suggested in comments to blow gently into your glass then swirl again to disperse the ethanol. Practice the art of the tease. Sip it sloooooowly. Josh Manson's advice to sip is very wise. You will still get a little bit of burn from the alcohol and its accompanying smell and taste, but it will be like a trumpet playing in the background of a smooth jazz session rather than a trumpet blaring into your ear. Quality counts. As Marco Cantele pointed out, get a good quality whiskey, not something that sells for $15 in a 750ml bottle. Jack Daniels or Wild Turkey are fine if you want to mix them with Coke, but not for this project. You don't have to splash your cash on a $90 bottle of single malt, but aim for something over $30. If you can't afford that, go to a bar that has a decent selection and try one glass. Get a reputation check. Ask what other people like. Ask the person in the liquor store or the bartender for a recommendation. Maturity counts. Whiskey changes as it ages, mellowing as the chemical compounds do their magic inside the barrel. People change too. If you're under 30, whiskey may be too wild for your taste buds. That's totally OK, come back in a few years and try again.Plenty of fish in the sea. (OK, fish may not be the best metaphor to use when talking about whiskey. Although grilled salmon is pretty darned nice with bourbon.) Anyway, if you get a whiskey that is just plain horrid, try and figure out what specifically rubs you the wrong way--bitter, sweet, tastes like medicine, tastes like grass--and move on to something else. Again, ask for a recommendation and say, "I didn't like X whiskey because it had too much of a Y flavor." Someone who's tried a lot of whiskey can help you avoid running into that problem again.
Some whiskey suggestions...
Starting with something smooth is a good way to introduce yourself to whiskey. Irish blends such as Clontarf or Tullamore Dew are very friendly and easy to drink, while still giving you something interesting to think about.
If you'd like to try bourbon, Jefferson's or Eagle Rare are both smooth and have a lot going on in the glass.
If you like sweet flavors, try a rye such as Bulleit or maybe Redemption, both of which are easygoing but have enough character to keep your tongue engaged.
Save the single malt scotch for later because there are so many variations that it's better to have an idea of what you enjoy before you start trying them... unless you're in a pub in Scotland, in which case you should definitely try several different ones. (Some people may hate my suggestions. Others will probably have their own suggestions. That's fine. Drink what you like. Life's too short to waste arguing over a brand name).
EDIT: Adey Hill suggested in the comments, "if you want to to try single malt scotch - start with something gentle such as a speyside, they are a great intro to the more serious malts."
Now get your butt off Quora and find yourself some whiskey.