Do it on the side.
Come up with an idea, work on it during your free time in a stack you want to learn about / would be beneficial to your career. When you get far enough, put it on your resume and see who bites.
I've worked for several large companies from local > 700 employees to national banks (as well as several < 10 employee companies). The side projects on my resume have had as much if not more significance then my corporate work. My current job hired me specifically for my experience with Rackspace hosting, which I only ever used on side projects. Point being, you never know what a company values until you look and apply.
Deciding what to learn might depend on what kind of job / career you want. Big corporate environments tend to standardize on Windows / C# / Java / Oracle / MS SQL, etc... while smaller companies / startups might be more open to trying newer things. Doesn't hurt to learn a few languages, but be careful of spreading yourself to thin. Pick a language or 2, do a few projects that employers can view and see where it goes.