Silverlight programming is more like using WPF or WIndows Forms.
the web browser sends your code to the client and that code is running on the client's desktop OS with a local copy of the Silverlight Runtime.
so like a desktop app you can have classes that can hang around for the whole time the app is running.
you can create "root level" properties in your app class if you need to have a global variable.
you can add properties to UserControls to pass data also.
the app can use a set of controls / classes that are found in the silverlight toolkit - that you download seperately
the toolkit includes a "Navigation framework" that treats UserCOntrols as child forms that are loaded into the main form's content area.
this is in some ways like a master page but in other ways is very different.
the bottom line is the Navigation control makes it easy to structure the app into logical pages and not have to re-load the browser page all the time.
you call WCF services or ASMX serives for your data connection to the server
OR
if you need a more special data service you can create a TCP socket service and call that.
think of this as "Flash for .Net" in many ways.
like flash developers use the flash IDE to create animations you can use Expression Blend to create SToryboards to create animation effects to make things look really good.
you can do animation form code also but Blend can let you see the animation and adjust it w/o writing any code.
you should go to www.silverlight.net and check out the getting started page and the boards there also there is a link on the getting started to a sample / walkthru by scott gu
also look at RIA services - still in pre-beta but they are IMHO a great addon to make building apps even simpler and more powerfull.
there is a kind of long video for them that show you a small demo app using them.
right now RIA is hard to get hosted as it is still not done, but on a local deveopment box you can run it and get started.
they should be final by the end of the year when VS 2010 releases from what i have been reading.
hope that helps.