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First and foremost, since it's web design...UI is the first place to test; icons, links, color, buttons, menu etc. then comes Usability; what is the flow, how will the user interact etc. and lastly Functionality; did the button works as design or programmed, did the menu show properly when clicked. These are just few by the way...but a good headstart for testing.
In the First Test HTML Tags to be Clean, CSS, JS, Jquery & etc... then go to make compatibility browsers test, then compatibility devices, then check the UI by sending the link for several users as i prefer Q.A peoples... fix the notes and bugs, compress & minify your files, upload it and enjoy...
I agree with Mr. Clodelio Delfino.
The UI is the first issue to test and make sure it is working fine and linking all other pages. Then comes Usability and Functionality of the site as a whole.