<p>Let's say I have a search page called Search.aspx that takes a search string as a url parameter ala Google (e.g. Search.aspx?q=This+is+my+search+string).</p>

<p>Currently, I have an asp:TextBox and an asp:Button on my page.  I'm handling the button's OnClick event and redirecting in the codebehind file to Search.aspx?q=

<p>What about with ASP.NET MVC when you don't have a codebehind to redirect with?  Would you create a GET form element instead that would post to Search.aspx?  Or would you handle the redirect in some other manner (e.g. jQuery event attached to the button)?</p>

You need to understand that MVC doesn't directly reference .aspx pages like WebForms in its URLs. Its main purpose is to separate concerns, that is model (data), controller (logic), and view (presentation).
First, you'd have to create a route matching your URLs, which would now look like this for example : /home/search/This+is+my+search+string
This would call the Search action method of the Home controller, which would get "This is my search string" as an input parameter. This action is responsible for accessing the model and pulling the results probably from a database.
Typically, your search action would then return a ViewResult containing the view placed in the folder /Views/Home/Search.aspx. Here, you can use neither the Postback functionality nor the events of your Web controls like in WebForms, because MVC applications are stateless and not event-driven. It's more like a request/dispatch way of doing things.
Read more about MVC here
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