Don't go to The Peel, its a shithole.
The others are good though.
Do you/your friend like sport? If so go to the Melbourne Cricket Ground (we just call it the MCG or just 'the G') and see an AFL game, its an awesome experience, especially if its a night game.
Chapel Street in the south is cool, as is Lygon Street in the North.
Great Ocean Road is fab but you would need a car and a spare full day.
Arbory Bar is a cool bar they converted out of an old rail track which overlooks the Yarra, really pretty at dusk/night.
Brunswick Street in Fitzroy is full of cool restaurants/bars including Veggie Bar and Smith and Daughters, two awesome vegan places, similar to Sydney Road in Brunswick.
Message if you want more suggestions.
Enjoy Melbourne! It's the best city in Australia!
), Brunswick, Prahan (Commercial Road) and gay bars/clubs over the weekend (The Peel, Poof Doof, Bottom End). 
