I'll try...
Places to eat:
a) If you feel like having your afternoon tea (or coffee) with a great choice of excellent home-made cakes in a 70s/80s environment I would strongly recommend this place:
http://gnosa.de/ Café Gnosa is frequented by gays/lesbians (as many other places in the "Lange Reihe", the street where the café is located), but, of course, everybody is welcome.
b) The "Turnhalle St. Georg" -
http://www.turnhalle.com/ - offers a great brunch (all you can eat, including coffee, tea & softdrings for 17.80 euros) on Sundays (booking recommended).
c) Fish market - every Sunday 5 am - 9.30 am (which is the reason why I have never made it there so far

).
d) Tower Bar -
http://www.hotel-hafen-hamburg.de/landungsbruecken_tower_bar_cocktail.aspx - have a cocktail and enjoy the fantastic view (booking a table at the window might be an option). The Tower Bar is close to the Portuguese quarter with many little restaurants offering great Portuguese food.
Places to see / things to do:
a) a boat trip through the city, including the beautiful "Speicherstadt". You can do a longer trip including the harbour if you are interested in big container ships & the history of the harbour, which is one of the biggest harbours in Europe.
b)
http://www.miniatur-wunderland.de/ - absolutely amazing (miniature landscapes showing various places from all around the world; very popular, so booking a ticket in advance might be an option to avoid queues)
c) Museum of Spices:
http://www.spicys.de/
d) If you like funny things, don't miss out on this shop:
http://www.lockengeloet.com/ Even better, the shop is located in St. Pauli where you can find lots of pubs, places to eat etc.
e) Depending on when you go you might have a chance to watch a match of St. Pauli:
http://www.fcstpauli.com/
In general, I would recommend just to stroll through the city centre - Hamburg is really beautiful and worth a visit. Enjoy!