How can I get a copy of the SQL generated by Management Studio?
Modern versions of SSMS (2008+ at least) allow you to generate a change script once you have made changes to a table using the graphical designer. You go to the menu item Table Designer > Generate Change Script.



That said, I agree with Thomas. In the long term, you should be proficient at T-SQL so that you can write ALTER TABLE scripts without the help of the wizard / graphical designers. They often do things you don't want to do.