The basis for making alcohol is simple. Yeast eats sugar and makes alcohol and CO2. All alcohol starts out that way. Wine is grapes or other fruit whose natural sugars are eaten by yeast and alcohol and CO2 is made, beer is sugars derived from grain like barley, wheat, and others which, again, is eaten by yeast and converted to alcohol and CO2.
There are more specific details...the yeast can only eat so much sugar...too much to start with will harm it. Yeast can only survive so much alcohol...too much kills it.
Wine is simply grape (or other fruit) juice that has yeast added and it ferments (the term for the yeast eating the sugars) over time. Once upon a time, the naturally occurring yeast in the air which would accumulate on the grape skins were sufficient to start the fermentation. Results were pretty random...sometimes it would be a good year, other times, not so good. Nowadays, winemakers kill the wild yeasts and add their own so they get consistent and predictable results.
Beer is similar, but there's an extra step. It comes from grain, but grain doesn't have naturally occurring sugars in abundance. When grain (particularly barley and corn) begins to sprout, enzymes develop. These enzymes convert starch to sugar. People learned to partially sprout and then halt the sprouting process to harness these enzymes, called "malting." The malted grain is then used to convert the starches into fermentable sugars...yeast is added and fermentation occurs.
Vodka and other spirits require one more step...distillation. Fermentation is the only way alcohol is MADE. Distillation doesn't make alcohol, it just concentrates it. To make spirits, you have to first ferment some kind of alcoholic brew like beer or wine or similar (in distilling terms this is referred to as "wash"). Once the wash has been brewed, it's put into a still where the alcohol is concentrated...it boils into a vapor, the vapor is collected and condensed back into a liquid...strong alcohol.
The brewing part is pretty labor intensive, whether you're making wine or beer. However it's perfectly legal to do so at home. You can either buy kits or start from scratch. Either way, you're allowed to brew up to 100 gallons per person of legal drinking age in your household, up to 2 people. The same applies to wine and beer (100 gallons of each per person...).
Distilling, due to safety and tax reasons is illegal to do in the home (ever) and not legal to do without a permit. The exception to this is New Zealand, Greece, and a handful of other countries that I can't recall at the moment.
Do a google search for home brewing supply stores in your area to be ablt to get the supplies you need and to talk to somebody who has tons of answers and will be very helpful for advice, recipes, and anything else you might want to know.