Properly stocking a home bar can be an expensive proposition. There are many different products you likely will want to buy, and a lot of them aren't cheap. The good news is, you probably won't be going through the bottles that quickly, and most alcohol stays good for a very long time.*
There are three different major styles of gin: London Dry, Old Tom and Genever. (You could also throw Plymouth Gin into the mix.) The good news is you only need to buy one bottle to start: London Dry. Tanqueray has long been my favorite, but Bombay Sapphire and Beefeater are excellent as well. Buy whichever one is cheapest.
It's easy to spend more money on vodka then you need to, especially if you reach for the Grey Goose because you "heard it's the best." By all means, pick up an expensive bottle if you're feeling flush. My favorite, which is medium-priced, is Stolichnaya. But you'll get by very well with some Sobieski.
Rum is a little more challenging, because rums vary a lot depending on what country they're from, what color they are (light/white vs. gold/dark), how long they're aged, etc. I would recommend starting out with two bottles, one of white rum and one of gold rum. Cruzan (from the U.S. Virgin Islands) is recommended — both cheaper and better than the ubiquitous Bacardi. But if you can find Flor de Caña (from Nicaragua) it only costs a little more and is excellent. If you don't drink much rum and only want to buy one bottle, I suggest you get some Appleton V/X, a very versatile and tasty rum.
Tequila has grown enormously in popularity over the past several years, which means there are now a lot of great choices on the shelves, in all kinds of prices. If your goal is to make Margaritas and other similar drinks, you'll want a silver tequila. I recommend either Camarena or Milagro. They're both affordable and easy to find.
This is a tough one, because there are so many types and so many choices. Do you go with a Scotch, Canadian or Irish? Bourbon, rye or Tennessee? If I were buying just one type, I would probably go with bourbon, and would probably get Maker's Mark. Maker's isn't the favorite whiskey of a lot of people, but it's a very good one and it's something that almost any whiskey drinker will drink without complaining. If you want to branch out and add a Scotch, I'd go with Johnnie Walker Black. Again, not always a favorite, but a crowd pleaser.
Cognac (which is brandy made according to certain rules in a particular area of France) was hot a decade or so back when the hip hop community discovered it, and brands like Hennessy and Remy Martin were name-checked in rap songs. It's cooled off since then, so there are plenty of good bargains to be found. (And also plenty of bottles that will cost you as much as a nice vacation.) If you want a simple brandy, I find Raynal to be quite good. It works fine in a lot of cocktails and won't set you back much at all. If you're looking for something a little more sophisticated, go for one of the cognacs made by Pierre Ferrand. (Their Ambre is very good and only costs around $40.)
If you're going to make any kind of cocktails, you're going to need some modifiers, with the most common being an orange liqueur. It might be triple sec or Curacao, but in order to make a Margarita or a Sidecar or Mai Tai, you're going to need something. There are many different types of orange liqueur, ranging from cheap to expensive. Unfortunately, the cheap stuff is usually not very good. On the upside, a bottle will last a long time, so it doesn't hurt as much to splurge. If you want a dryer liqueur, go with Cointreau. If you want a sweeter one, go with Grand Marnier. Yes, they're expensive. But they're so good that you'll be glad you spent the extra money.
If you're planning to make Martinis or Manhattans, you'll need to get some vermouth. Sweet (red) vermouth goes in a Manhattan and dry (white) vermouth goes in a Martini. There are some high-end brands that are delicious. But on the affordable end of things Martini (sweet) and Noilly Pratt (dry) work very well.
You're not going to get this done without spending a couple hundred bucks. But once you do, you'll be able to make a lot of drinks — and save yourself a ton of dough over what you'd spend in a bar. Plus, with a little practice, you'll be able to whip up some great cocktails that will quickly make you the envy of all your friends and neighbors.
*Except for vermouth. Vermouth is only good for a couple of months once you open it. And only if you keep it in the refrigerator. And yes, I know vermouth isn't a spirit. Neither is orange liqueur.