If you do choose to get a stick, what you don't want is a cheap one. The first time you use a stick, there's a 99% chance that you will be worse at execution than you were on a pad. The 1% exception is if you were for some reason playing a 2d charge based character, where you will most likely see an instant improvement.
Playing on a cheap stick just feels bad, and when you're executing even worse, you're not going to like the experience, period. Sticks take time to adjust to, and a lot of people have trouble with the square gate common to most sticks as well. A cheap stick will just feel sluggish, unresponsive, and not worth practicing on.
The best options would be to get a commercial stick with real arcade parts, like the Tournament Edition sticks from madcatz, an easily moddable stick like the Hori Real Arcade Pro (only need to change out the buttons), or get someone else to make/mod a stick for you (well, could be you if you're mechanically inclined). Expect any option to run you around a 100 dollars, unless you mod a really cheap stick or build one yourself.
As for whether or not sticks are better...the best controller is the one you're practiced in. In SC, you're not at a real disadvantage if you don't have a stick to practice on. 2d fighters, on the other hand, are just flat out painful to play on a pad, and it's easier to push the limits of execution if you practice on a stick (a lot of this is because 2d fighters tend to have more than 4 main buttons). You can still be good at them with a pad, especially when multiple button mappings are allowed.
