Skype has done fine work with its new Version 4.0 on Android, bringing about some badly needed performance improvements as well as a redesign. But in doing so, it's buried the sign-out button, as well as the settings — two things that absolutely need to be in the top level of the user interface.
Here's how to get to them:
When you're logged in, tap the image of your face (or whatever you're using as an avatar) in the top right corner. That'll take you to your account profile. Then hit the overflow button — the three dots — in the lower right corner, or hit the menu button if you're on one of Samsung's phones (or any other with an actual menu button). Now you'll have the option to sign out, or to go to the settings.
It's worth noting that if you want to log in with a different username, you'll need to make sure "Sign in automatically" is unchecked in the settings menu.