You could add an event handler to your search button and do something similar to this
private void button_Click(object sender, EventArgs e)
{
String query = queryTextBox.Text;
Response.Redirect("SearchResults.aspx?query=" + query);
}
Using JavaScript
function doSearch()
{
// Assuming you are not using jQuery,
// using jQuery it would be $('#queryTextBox').value instead
var queryString = document.getElementById('queryTextBox').value;
window.open("SearchResults.aspx?query=" + queryString);
return false;
}
Html
<input type="text" id="queryTextBox" />
<input type="button" onclick="return doSearch()" value="Go" />