I've never been too keen on alcoholic beverages since trying things growing up and not liking them. However, I try to keep an open mind and would like to taste more. My problem is I'm hesitant to buy my own packs of beer or bottles of wine or drinks when I'm out, as that might get expensive. Is there a way to taste a variety of beverages without breaking the bank or sampling off friends' drinks all the time?
As far as beer goes: Many brewpubs will offer a sampler (or flight) of beer that includes small (few oz) glasses of many of their beers from IPA to stouts and wheat beers. You get a few sips of each beer for the price of a pint (or so) and you can determine what you like or don't like. Also, sometimes brewpubs offer samples of their beers if you ask. I went to a place offering habanero beer, and definitely wanted a sip before buying a pint. It's not always advertised but worth asking if you have an interested in something in particular.
For wine: Many places that sell wine also offer tastings on occasion, also wineries generally have tastings if there are any in your area. Wine stores usually will have a red and a white available during tastings and the person in charge of the tasting can recommend what kind of food to pair it with. Tastings are a great way to get to know what kind of wine you like!
For liquors: Finding a local distillery can be tricky if there are none where you live, but I'm lucky to know of a few near me that have tastings for a small fee. A caveat, though, liquor can be hard to drink for some without being mixed with some sort of mixer (juice, soda, etc). So just keep that in mind, when you sip some gin, you may enjoy it mixed with tonic even though you dislike it on its own.
If you are not keen on the taste of alcohol, I would start with cocktails.
Go to a good cocktail bar (ask for recommendations from friends or websearch), take the list and pick drinks that are mixed with things you like. Make a night of it! Try and go with a few friends and all order different things. If the bar tender is good he will help you out with good selections etc.
As for beer - Lemontwist has a good idea. Any good Ale pub here in the UK will offer you a sample before you buy. Also many brewers also produce a "variety box" which contains 1 or 2 bottles of approx 6 different beers (usually 9-12 bottles total).
Wine - Lemontwists idea is again a good one. But wine is probably the toughest thing to try, trying the same wine with certain foods, at certain temperatures or having breathed a different amount will greatly affect it. I'd either try the systematic (pick a price point, buy a glass of wine, note its details and your rating, lather rinse repeat (Im sure there is an app for that)), or just go for luck - pick at random and try and remember the ones you like.
Find a local bar with a cheap happy hour and get friendly with the bartender. If they're any good, they should be well-equipped to help you find a drink you'll love.
Side note: I was going to suggest myopenbar on the off chance you lived in NYC, but it looks like they just recently closed up shop :/