Wait

Wait = stay somewhere until someone comes: ‘I’ll wait here until you get back.’ ‘It’s quicker to walk than wait for a bus.’

Expect = believe that someone or something is going to come: ‘The train is expected to arrive in the next five minutes.’ ‘I can’t leave the house. I’m expecting visitors.’
Wait for sb/sth: ‘I’ll wait for you outside the post office.’ ‘What can I do while I’m waiting for the paint to dry?’

In informal styles, to show that you are really looking forward to something, use I (just) can’t wait or I can hardly wait: ‘I can’t wait to see you again.’ ‘I can hardly wait for the holidays to begin.’