I agree with @logophobe; open-ended and/or subjective questions are frowned-upon as you probably know from your other SE sites. That said, I like mixology (though I despise that term...), but I am no professional. I'll try to pick off a few parts that were helpful for me as I started this hobby, and hope that will either help you get started, or help you refine your question! I tried to arrange this answer from specific to general, because I couldn't stop typing. :)
The tl;dr answer: pick one drink. What's your favorite drink? Buy the ingredients. Mix it straight from the book. Taste (hopefully, enjoy). Vary the proportions. What happens? Tweak. Then pick another one; repeat. Alternatively: pick one base spirit; riff on that.
The longer answer... You articulate the universal problem in mixing: how to begin without going crazy or broke (or both). My ridiculous answers below also illustrate this: you can spend unbounded time thinking about this and talking about it. :)
Here are some examples of how I'd suggest starting, because, apparently, I like hearing myself type:
- Pick a booze. E.g., vodka: first, sip it by itself. Then mix with endless variety of mixers or juices: tonic, orange, cranberry, grapefruit. Explore mixers: triple sec, peach schnapps, Rose's lime juice. With that alone you can make at least a dozen legitimate cocktails (cosmo, cape codder, SotB, greyhound, fuzzy navel, ...).
- Pick a drink. E.g., manhattan: start with a reasonably inexpensive bourbon or rye whiskey that you like, some decent sweet vermouth, Angostura bitters. Mix by the book. Vary the vermouth: half-and-half with dry vermouth: perfect manhattan. Vary the bitters; see which you prefer (citrus, Peychaud's, Angostura). Drop the vermouth and add some sugar: old fashioned. Add anisette liquor: whiskey Sazerac. A similar riff could be done with many other classic cocktails: e.g., dry gin martini, pink gin, Americano, Negroni, ...
Edit: adding response to Mojito/Rum preference. Given that you like the Mojito (muddle lime, mint, and sugar; add rum; fill soda) and rum; I'd start like this:
- Back to basics. Having very similar (though simpler) ingredients: Try a daiquiri. First, this gives you an opportunity to learn how to make yourself some simple syrup -- sugar syrup; recipe. Make a bunch and keep it (once cool!) in the fridge. Ice in a glass, add rum, lime juice, then some of your brilliant simple syrup. Shake, strain (or leave on the rocks, my preference), enjoy. This also gives you the opportunity to try a cocktail shaker with pint glass and a cocktail strainer.
- Add a new ingredient. Mix it up a bit with some grenadine; you get a Bacardi cocktail. This gives you an excuse to buy (A SMALL BOTTLE!) of Grenadine. (You see where I'm going with this...)
- Now go back to the Mojito. Get yourself a wooden muddler and mash up the solids first, right in the glass: muddle the (quartered) lime, mint leaves, sugar, until crushed. Then add your liquids and stir.
- Garnish! Add a straw-shaped slice of some fresh sugar cane stick and fresh mint sprig on the top. Presentation!
- Change the spirit. If you like white/light rum, look around for cachaça. It's a relative of rum (both made from sugar cane; rum from refined/molasses products). See if you like it straight or chilled, then try a caipirinha. Try cachaça in place of rum and vice versa. I find the stuff rather vile, but it's a unique taste.
Anyway. I hope you see what I'm trying to illustrate: just change one dimension at a time. This gives you the opportunity to see exactly how each feature (spirit, garnish, mixing technique, gadget, flavouring) impacts the result. Just my take. Have fun!
Then, more generally: Mixing involves three categories (from my perspective):
- ingredients (basically, dictated by what cocktails you're trying to make)
- technique (i.e., how to make them)
- gear (i.e., what you need to make and serve them; like measures, glassware, shaker, utensils)
If you want to get really fancy, any of these could (figuratively) consume your entire life and income. To start, you need only a few bottles and some ice. As you say, collect more over time.
Since it sounds like you're focused on the ingredients (which, IMHO, is the right place to start) my basic recommendations are as follows:
- Start by exploring a few classic cocktails that have a small number of basic ingredients. Consider looking at
- Understand the classes of ingredients and what they are:
- base spirit (e.g., vodka, gin, rum, whisk[e]y, tequila, brandy),
- liquors (e.g., triple sec, anisette, coffee liquor, herbal liquors),
- other additives or garnish (e.g., bitters, fruit, herbs)
- mixer (e.g., sour mix, simple syrup, soda water, tonic, fruit juice, milk/cream)
- Mix what you enjoy! Could be boozy classic cocktails, fruity vodka drinks, easy-going aperitifs, ...
- Get a cocktail book; read :)
Because I can't help myself to give a general answer as well: A bunch of sites and blogs have recommendations for general bar ingredients; search for something like "essential cocktail ingredients" to get started. I like the following ones (beware ad content...)
Last, my overarching recommendations: Start slowly. Taste everything. Have fun.
Hope it helps! Enjoy (responsibly). :)