Yes, yes, yes, yes! Being able to track, prioritize, and manage your issues is key to successful software development. With one person, you can (almost) get away with a spreadsheet and zipping up old source trees. Adding even one developer to a project changes things dramatically -- suddenly, issue tracking and source code control is necessary, or you're going to be dropping issues, overwriting functionality, and generally having a miserable time of it.

I'm surprised that no one's mentioned StackExchange's parent company, FogCreek, yet -- their FogBugz software is the best issue tracking app I've ever used. High speed, low drag, and affordable, especially if you use their hosted solution. They used to have a free hosted trial that had, I believe, two user licenses free -- that may not be the case anymore, but I'd recommend checking it out.