Uncle Lloyd's Tips and Tricks:
It's hard to simulate hops and barley, so making fake beer is more a matter of suggestion. Root beer and ginger ale don't taste like beer – but they are brown, fizzy, and have the word "beer" or "ale" in them, so that's often enough to convey the message of beer. Cream soda is also a decent choice, since its creaminess suggests the creaminess of things like stouts.
(Consider also mixing the sodas above, so that it's not immediately obvious to the taster exactly what you've put in. This will make it seem less like "a cup of ginger ale" and more "a cup of some soda that's meant to be beer".)
Beer does have a bitter flavor by nature, and very few other beverages do. One idea is to add a little Moxie, a New England soda that to me tastes (in the words of Ryan Cunningham) like old tires, but which some folks seem to love. Dig around at grocery stores or beverage stores and you should be able to find it.
Of course, if you're feeling wealthy, just go with store-brand non-alcoholic beer. Caliber is my personal favorite.
A grape juice base is obvious. Then consider adding other red or purple juices to it, e.g. the cran-grape mixes you can find easily in stores. For port, add extra sugar, or buy juice concentrate and don't add enough water.
If your budget allows, you can buy non-alcoholic wines at some stores, too. These are essentially grape juice, but less sweet.
The one thing that all liquors have in common is a strong kick from the taste of the alcohol. Your goal is to simulate that my having something curious or strong in the flavor: extra sweet, extra bitter, etc. This could mean adding lots of sugar, honey, sour mix, ginger, and the like; it could also mean using a recipe that's got a kick to it already.
First, start with your base. Go with the suggestions below, or use your own. As with beers (above), consider mixing multiple flavors to get an unfamiliar result. For instance, try mixing fruit juice with those weird flavors of Gatorade, or mix iced tea and ginger ale.
Whiskey (and scotch, etc) is made from distilling various grains, then aging them in wood. There isn't much else that tastes like this, so you have to wing it.
Brandy is fortified wine, so it should start with wine recipe above. Then add a "kick" – probably extra-sweet – and then some drops of brandy flavoring, found in the cooking supplies aisle of any grocery store.
Schnapps should start with the flavor in question. Peach schapps, obviously, should use peach juice or the like. Since schapps is traditionally sweet, go for the extra-sweet "kick" by adding lots of sugar or honey.
Vodka, properly done, tastes like alcoholic water. Good luck simulating that! Consider starting with tonic water (which doesn't taste like vodka, but the quinine has a "kick" to it) and letting it go flat, and then adding something to it to strengthen the kick. Salt? Anise flavoring? Vanilla? You be the judge.
Sake is another drink that's not easy to replicate. Under the theory that the sake sold to Hesketines would be the cheap, low-grade crap, the sake I made for Ten Thousand Daggers was essentially water, with a little anise flavoring and a bucketload of salt in it.
Gin isn't especially period (it's from the 17th-century), nor is rum (which came from the Caribbean). If I felt like being anachronistic, I'd start with flat tonic water and cola for these, since we mentally associate those sodas with the liquors in question. Then I'd add some flavor to the tonic water (maybe a little grapefruit juice and a pinch of cardamom?), and stir brown sugar and water into the cola.
Due to the dwarven constitution and the agricultural options that dwarves have available, all bets are off with dwarven brews, and they don't have to try and taste "familiar". We've made dwarven stouts with pudding mix (adding extra milk). Other dwarven drinks have essentially been watery Jell-O. Dwarven whiskey is traditionally bright green, to reemphasize that it's poisonous to humans; this is made by mixing blue and yellow Gatorade and then adding lemon juice.
For games that are long or during hot weather, be careful with your ingredients. Many substances say "refrigerate after opening" for good reason.