I just have a lot of hooch built up over some time, Clare, everyone's welcome to an opinion. I don't want to give the impression I am an oracle!!
OP, what I think I've gathered over time is to build up slow - you don't want to freak out and think you have to be able to make every cocktail people ask for at once. I think out of the sauceguide on the bottom shelf I can do 2/3 to 3/4 of a page opened randomly, so it's impossible to think you can ever be more than a pretty workable home bar.
You have 2 options, really, and it depends on your personality and houseguests.
If you like your drink, and want to enjoy the stuff yourself, I'd recommend you expand your bar in avenues you appreciate best. So, for me, it was always whisky (Scotch and Bourbon) and gins (Plymouth, Blackwood) that really interested me the most. Then I heard of recipes that had the spirit I like in, and added mixers to help those along (so, e.g. for Bourbon, the "Currier" had lime and Kummel in, so I added Kummel recently. Old Fashioneds and Sazeracs needed bitters, and the latter also needs Absinthe and Cognac, so I built up in those directions too).
That method plays to your strengths - but does restrict your range if you want to have a party.
If you're more of a host (this is definitely me, I almost never touch the bar unless there's a party on, and I spend most of my time each night in the kitchen mixing things for others and just tasting them as I go out of interest) then the best thing to do is hit up key ingredients which allow you to make a vast range of drinks. Again, the sauceguide I have lists a bunch of these, which I think you can actually refine further if you wanted. They list these, and the ones I put in italics I think you can move down the list. the Bolds I'd prioritise:
Spirits:
Vodka (I am not a big vodka drinker so will defer to those who recommend a specific brand, but it's an absolute essential)
Gin (Plymouth is a good starter gin. If you get a taste for it the range of botanicals can really be broad, so move into Hendricks or small still like Sipsmiths, maybe Millers. Most classic cocktails are big on vodka, gin and cognac.)
Rum (white is a pretty good standard, again I will defer to others on brand as I sometimes just buy own brand. I like to buy decent Cachaca, or Sugar Cane Spirit like CSR, rather than spend more on rum).
Bourbon (In my opinion you're wasting time mixing most good single malts, they can be very dominant. Bourbon has a great standalone flavour AND mixes really, really well, short or long, it's the must have whisk(e)y on the bar. Saying that living in Scotland can be risky business!).
Cognac (I'd consider this very important as a lot of classic cocktails rely on cognac. You won't find yourself using it all the time but you can make some real crowdpleasers with cognac. It also mixes well with e.g. bourbon for Sazeracs, which are the a-bombs of the cocktail world - strong and delicious).
Tequila (I would consider this essential as I love margaritas and spins on other classic cocktails with Tequila. Drank some good stuff with Guruman off SSSH. However, if you aren't a Tequila fan, I wouldn't say you need to have a bottle to keep a party going! Also it reduces the number of people who imo waste it on shots).
a blended scotch (J&B or a simlar will do, I'd avoid Bells and Grouse unless you have no choice. Imo bourbon is the clutch one). If you're going down the whisky route it's a matter of time before you get irish, scotch, tennessee and bourbon all on the same bar!
Liqueurs:
Dry Vermouth (later you can get Rosso. Dry first. A gamechanger in the cocktail world and needed for the dry martini and a billion others. Your first liqueur requirement, closely follwed by Cointreau).
Cointreau (absolute essenial. It can double as triple sec and replace Grand Marnier - but isn't the same, GM is brandy based, Cointreau is closer to eau de vie. But in a heads up battle, you NEED Cointreau on your bar. It's a staple liqueur of so many you're mad not to have it.)
Chambord (this is black raspberry liqueur, but will work to replace most cremes you'd need - strawberry, raspberry, if you need a fruit liqueur Chambord will do it.)
Also, very importantly - French Martinis are a blast with the ladies (2s vodka, 1/2s Chambord, 2 1/2 shots pineapple juice, shake with ice, serve 'up' (in a cocktail glass), garnish with 3 raspberries, the ladies will go wild, get wasted and love you forever.) You're welcome.
Grand Marnier (Useful but if you've got Cointreau, wait till later)
Kahlua (or Tia Maria, good for White Russians but really unless you have coffee fiends you can add this anon as well).
Apricot Brandy (opens up other avenues of cocktails but not essential to begin)
Mixers:
Champagne (or Cava, or Prosecco. Yes, I know it's technically wine but you usually add stuff to it, be it peach puree (bellini), a bitters soaked sugar cube (classic champagne cocktail) or something crazy like the B2C2 (Brandy, Benedictine, Cointreau and Champagne, invented in WW2 by US soldiers who found all this stuff as they liberated France, woo!!)).
All essential to boost your bar, listed in order of importance:
Juices: Orange, Cranberry, Pineapple, then Grapefruit and Apple.
Sodas: Soda Water, Cola, Ginger Ale, Ginger Beer, Tonic.
Syrups and Cordials: Lime Cordial, Sugar Syrup, Grenadine
Fruit and Veg: Mint, Lemons, Limes, Fruit (raspberries, strawberries, cherries)
Weird additions: cream, milk, eggs, Vanilla pods, sugar cubes, honey/maple syrup, coffee/tea.
and of course - ICE! Lots of ice.
Remember once you've passed down this route you can add flavoured vodkas, syrups, extra liqueurs and weirder ones, Pisco, amaretto, cherry brandy, Xante (pear liqueur), or other things you like like in my case Calvados. Really, the booze world is your oyster. But start with the essentials and build up. Would welcome a debate on brands!
To add something else interesting to the mix Mrs Harris found this recipe for Lillet Vermouth Buttermilk milkshakes. Interesting! I'm not a vermouth fan but will give it the old College try some time soon and report back!
Lillet Buttermilk Milkshakes (makes 4)
2 cups vanilla ice cream
1/3 cup Lillet/other vermouth
1/3 cup buttermilk (or milk if you can't source)
basil flowers/leaves, optional
Let the ice cream ripen. Add it to a bowl or milkshake cup.
Pour the Lillet and buttermilk/milk over ice cream; mix to combine by either maker, shaker or cream by hand.
Serve in topped with a few basil flowers/leaves if handy.