I have two classes in my game, a Main Menu and the main playing screen. How can I swap through classes? E.g I click on "New Game" in the menu class and it changes it too the Game class. A transition animation explanation would be nice.
You need some kind of gamestate-management system in place.
I could write a long answer explaining this, but people who are a lot more experienced with XNA and game programming than me have already done that, so I shall point you towards them.
Personally I've found this example to be excellent: http://create.msdn.com/en-US/education/catalog/sample/game_state_management
It shows a gamestate-management system with the ability to easily add as many gamestates as you like, as well as defining transitions in and out of the gamestate.
You should give it a read!
You probably want to look into state machines. Here's a popular App Hub tutorial on game state management, and here's another random Google result for XNA state machine information (it's not really a topic that needs to be specific to XNA, so you can get broader search results in Google if you exclude the "XNA" bit, if you want to research the topic more yourself).
Once you have the basic implementation of such a system, you can expand it by adding (for example) animations that play as you transition in or out of states.
Game State Management is what you're after. Once you have the states sorted out, when they transition you can create whatever effect you desire (fade out, in, black out, white out, etc) and that's what shaders are for. But that's probably a slightly separate question.
Another article that could be interesting is this which describes an actual application of it.