I can't tell you how to stock your bar. Ideally, really, the process of learning how to drink and to make your own drinks would start you off with a certain array of bottles and end with you going into the store and restocking along completely different lines, lines tailored to your tastes.
Maybe you love tiki drinks, for instance. Though tiki attracts collectors of elaborate drinking vessels and complicated garnishes and whatnot, the essence of tiki is the blending of fresh juice and flavored syrup with multiple rums. While that may sound simple -- while at first glance it may even sound dangerously close to the appletini/sour-mix culture we 21st century cocktail enthusiasts are trying to push away from -- the key is the multiple rums, as many as five different rums in a drink. Now, me, I very rarely make any tiki drinks at home -- because that $100+ buy-in just to have enough different rums on hand feels better spent on three liqueurs and a great sipping whiskey, or absinthe and three different bitters, or ... you see what I mean. It's a hefty layout, and my drinking interests go in another direction. I go for variety and flexibility, which oddly means avoiding the tiki option -- because it frees my budget up in so many other directions.
But I also never buy brandy. I'll order a sidecar sometimes in a bar, but for whatever reason, brandy drinks don't interest me much. This is a little odd given that, before becoming a fan of cocktails, I'd drunk gallons and gallons of Remy Martin in my life. But so it goes: I have come to realize I don't have much interest in brandy drinks, and so I don't stock brandy.
The base liquor I am second-least likely to have on hand is tequila, and this is pure economics. A tequila of quality X costs price Y*1.5 to Y*2, where Y is the price of a non-tequila base liquor of quality X. For the cost of a bottle of Patron, I can buy a bottle of Plymouth gin and a bottle of Bulleit bourbon. So, again -- it's hard to justify that tequila, harder still to explore different brands. Sometimes in the summer when I either anticipate making a lot of margaritas, or plan to infuse a tequila with something, I'll buy a bottle of Sauza Hornitos, which is fairly cheap and certainly as cheap as your tequila purchasing should ever go.
It's a shame how much of the liquor market is impacted, defined, by labeling laws. I suspect there would be a strong market for American-made agave distillates from craft distillers.
So. Let's start with ... 10 bottles. A 10 bottle bar. For me, that would be:
Bourbon. Probably Bulleit, which has a strong enough rye character to split the difference between bourbon and rye, and therefore works well in a Sazerac.
Gin. Usually Plymouth, Bombay, or Citadelle. I don't drink martinis unless I'm out.
Rum. I'm not faithful to any specific brand of rum. I tend to go for aged rums over clear rums. Prichard's is one of my favorites, especially for sipping, and Clement is very good. Pampero Aniversario, an aged rum from Venezuela, is the rum I buy the most often. The rums I currently have stocked are Prichard's, St Lucia Distillers Chairman's Reserve, and my kola-infused rum.
Genever. The only brand I've had is Boomsma's oude genever, but I can't imagine not having it or another genever in my bar now.
Luxardo Maraschino. There is really no substitute, brand-wise.
Aperol. Yes, I'm picking Aperol over Campari -- even though I can buy Campari locally but have to mail-order Aperol. The reason is pretty simple -- although Aperol works in every Campari drink (which is not to say it's always an improvement, but an Aperol Negroni, for instance, is a perfectly good drink), Campari doesn't really work in Aperol drinks like the Shaddock or the 2 to 2. Aperol is fantastic in combination with genever, for instance, leaving visible malty notes that Campari would overwhelm. So picking Aperol here simply gives me more options.
Punt e Mes. My vermouth of choice, though I know it barely counts as a vermouth.
Absinthe. There are only a handful of absinthe drinks that I drink with any regularity, but a bar in which I cannot make a Sazerac is no bar at all.
Angostura bitters. The best go-to bitters, though Fee's whiskey-barrel-aged -- which is actually what I have in stock right now -- are a good substitute.
Peychaud's bitters. Less versatile than Angostura, but necessary to make a Sazerac, and good in Trinidad Sour type drinks.
Highlights of drinks you can make with this bar: Old-Fashioneds or Sours with four different base liquors, Sazerac, Punt e Mes Bulleit Manhattan, Aviation, Daiquiri, Aperol Negroni and Boulevardier, modified Shaddock (normally equal parts genever, Aperol, St Germain, and lemon, you would need to double the Aperol or sub Maraschino for the St Germain), various drinks made on the Trinidad Sour template, 2 to 2 (1.5 oz Aperol, 1 oz absinthe, 1 oz lemon, 1/4 oz simple, bitters). (While we're modifying the Shaddock, I actually usually make it with 2 parts genever, 1 part each of the other elements, so our 10-bottle Modified Shaddock would be 1 oz genever, 1 oz Aperol, 1/2 oz lemon juice, which I suspect is a terrific drink.)
A 20-bottle bar? Add these:
Rye whiskey. I know it's heresy to put it in the 20 instead of the 10, but I haven't got many rye whiskey options: Old Overholt and Jim Beam year round, and a few at the $50 price point at Christmas. Now, the Papa Saz currently in my bar is probably the best-tasting liquor I own, but I couldn't afford to make it a mixing staple, nor do I have year-round access to it anyway.
Tequila. It's not that tequila is a bad base liquor in the slightest, of course. It's kicked down to the 20 for the economic reasons I already mentioned.
Corn whiskey. The reason I don't have this in the 10 is because I had to include both absinthe and Peychaud's bitters for the sake of Sazeracs. I'm a proponent of unaged corn whiskey and its cocktail possibilities, and am working on a post about exactly that.
JDT. My own thing, Jack Daniels infused with Luzianne black tea. The JD is so mild and mellow that it gives you a good space in which to put those tannins. I nearly put this in the ten but couldn't find anything to take out.
Clear Creek douglas fir eau de vie. There's just nothing else like this, and yet when it works in a cocktail, it really works -- and the taste straight is amazing.
Ransom Old Tom gin. Post forthcoming about Ransom, which is a midway point between genevers like Boomsma and light London-style Old Toms like Haymans.
Campari. So weird to have that on this list instead of the other. Of course, I say that about almost everything on this list, don't I? Campari is the best of the amaros, even if you don't consider it an amaro. Like gin and tequila, it is also an exceptional shot to add to potable citrus juice -- orange, grapefruit, tangerine, etc -- and is useful in fruit gelatin preparations or maceration.
Green Chartreuse. I really only use this for The Last Word, but that's a cocktail you can make a million different ways.
St Germain elderflower liqueur. Bartender's ketchup, useful as the sweet element in anything built on a Last Word or Aviation template -- less assertive than maraschino or creme de violette, it blends with basically anything. (Though I find this is also true of Domaine de Canton ginger liqueur.)
Cynar. Cynar has a terrific affinity for both huckleberries and muskmelons, so in the summer I go through a lot of this mixed with gin and those juices.
The list of drinks made available by the 20-bottle bar is extensive and I'm nowhere near finished exploring it myself (I would guess I have about a 30-bottle bar myself, though it gets complicated when you figure in the various infusions -- huckleberry gin, coffee-cherry-orange liqueur, kola rum, white cassis, etc etc). But here's one:
2 oz Aperol
1 oz JDT
1 oz lemon juice
The JDT is easy to make: pour half a bottle of Jack Daniels over a family-size Luzianne teabag and keep an eye on it for the afternoon. You want it to be noticeably tannic. You're basically making sun tea with whiskey instead of water. The Aperol really plays well with those tannins, and makes this drink surprisingly similar to a lemonade iced tea/Arnold Palmer/half and half.