I am considering that you are using
VB.Net
Let the Event handler of the button click event be ..
sub button1_click
response.redirect("newpage.aspx?carrytext=" & textbox1.text)
end sub
where newpage is the name of your new file and carrytext is the name of the argument
Once in NewPage you can access carrytext using
request("carrtytext")
e.g. response.write(request("carrytext"))