How do I carbonate a drink that originally isn't carbonated without altering its taste or the process taking more than a few hours? I've seen some methods that use yeast, but that requires about two days and the taste changes, so that won't do.
Wilds of Russia? If you have dry ice, just adding pieces of it (chopped small) to your drink will work to some extent. Here's a vid: How To Make Soda With Dry Ice Be aware that not all dry ice is food grade. If no dry ice, you can get by with sodium bicarbonate (baking soda) and vinegar (acetic acid), or some other acid. Just mix the two chemicals, and bubble the resulting gas through your drink. Here's an Instructable, w video: Carbonating: The Cheap and Easy Way. It'd be easy to improve on the kid's method, for example by adding a bubbler stone, but the general principles are sound. Either method will cause some change in your liquid's taste, as you're lowering the pH by generating carbonic acid, but that's true of any carbonation scheme.
How to do it - practical advice
You need a machine for carbonation. There are different brands. You buy cartridges with gas for them, put the bottle with the drink in the machine with the loaded cartridge, and press a button a few times. The drink is then carbonated.
I there are also combined whippers/carbonators which can alternatively take a nitrous oxide cartridge to make foams. This is too complicated and unnecessary if all you need are carbonated drinks. If you are like most people out there, you want the simple system which just carbonates.
Update: Some more words on what forms of carbonation are available
Carbonation is a process in which you get CO2 bubbles dissolved in a liquid. The gas doesn't stay dissolved for a long time, it starts "seeping out" of the liquid, forming tiny bubbles, which rise through the liquid, giving a prickly feeling on your tongue when they reach it.
You can get the gas into the liquid by three general classes of methods: add an organism which produces CO2 (biological method), add a compound which reacts to produce CO2 (chemical method), or add the gas directly (physical method).
Manufacturers sometimes combine machines which can expel CO2 from cartridges with machines which can expel NO from cartridges. Expelling NO is used for whipping foams, e.g. creating whipped cream. The whipped cream spray bottles in the supermarket are a simplified version of such machines, which cannot be reloaded.
While the construction of CO2 expelling machines and NO expelling machines is similar enough that it makes sense to combine them in a single body, the use cases for CO2 use and NO use are very different. If you only want to carbonate, a CO2 machine is sufficient.