Upgrading speaker wire is probably one of the cheapest and easiest tweaks you can do. It’s generally accepted that larger speaker wire is better, especially if you’re starting with the cheapie 20 ga. stuff that comes with most all-in-one systems. As to how audible an improvement you will hear, that is largely dependant on your speakers and electronics. However, people seem hear a difference with larger-guage speaker wire more than any other wiring upgrade.
For instance, many years ago a good friend of mine wanted to upgrade his speaker wire from the skinny stuff to some plain 14 ga. zip cord, and he asked me if I thought he’d hear an improvement. Well, we were talking about some so-so speakers that came with a rack system (remember those from the 80s?), and short runs of only 6-8 ft. Plus, he didn’t really have a discerning ear, in my opinion. So I told him, “I doubt it.” I was really surprised when he reported back to me he could hear a definite improvement.
That said, Jason, if you upgrade, get the biggest wire your speaker terminals can handle – 12 qa. if possible. Some systems have a problem with wire that size; if so, go with some 14 ga. Doesn’t have to be anything fancy or expensive. Anything you can find at a hardware store will be fine.
It won't make much of a difference unless the wire is grossly inadequate. It's also not a big deal there are a lot of other tweaks to do than speaker wire that make much bigger differences, such as room acoustics. I'd put speaker wire pretty much last, just get decent zip cord for now, unless you have a very high-end system, in which case do room acoustics first, then wire stuff.