Infusing is adding flavor, and to a lesser extent aroma, mouthfeel, color, and some less easy to define character. As a result, when choosing your spirits, you want something clean, relatively good quality, but not so sublime that it would be criminal to monkey it up with other ingredients. Some general guidelines:
Vodka – Look for middle-shelf brands. Shmirnoff, Skyy, Svedka, Boru, Fris, Sobieski, and even one Kirkland store brand (based on a tip) have made it into my infusions with great success. Look for unflavored, at least triple distilled, and filtered. Generally, I pick up whatever’s on sale at my nearest warehouse style liquor store. Another option is using an active charcoal filter (like Brita) on a cheaper vodka. I avoid this for a few reasons–the filters and pitchers are expensive and can’t really be switched back to water use, you end up wasting whatever gets trapped in the filter (and then evaporates), the filters only work when wet, AND the results are still inferior to a basic Shmirnoff. More trouble than it’s worth.
Rum – Any OK clear rum will do. Flor de Cana from Nicaragua is a very very nice clear rum but it sometimes runs a little expensive in the US. Bacardi Silver is a very safe bet. Spiced, aged, and dark rums have already had a lot of flavors imparted in them, so it’s best to avoid them.
Whiskeys – Generally any quality brand of bourbon, scotch, rye, or other whiskey will be fine. Jack Daniels, Canadian Club, Dewars, Jim Beam–all will be OK. Personally, I find Canadian Club a very basic, simple, and smooth whiskey that is inexpensive. It lacks character, which is a good thing when infusing. Any single malt or high-end blend should be enjoyed on its own merits.
Gin – Gin is already so flavorful that I tend to prefer a light hand on the few infusions I think worth trying. In this case, buy whatever gin you’d happily enjoy a Gin & Tonic with and you should be fine. I really don’t like cheap gin so I tend to stick with Bombay and Tanqueray. I generally experiment with Gordon’s Gin as it has the best ratio of decent quality to cheap I’ve found.
Tequila – Again, I prefer a clear and clean tequila, though some exceptions can be made. Tequilas tend to run expensive. I’ve found Jose Cuervo Silver works rather well.
Others – For things like Cachaça, Pisco, Grappa, Brandy, etc., I generally play it by ear and will offer tips when they come up. Generally, anything you’d enjoy drinking in a not overwhelmingly sweet mixed drink will be more than good enough. Unflavored and clear are usually other traits to seek.