Lots of trains from Köln Hbf to Düsseldorf Hbf (no reservation req'd) -- from there you'll take the U-bahn in the direction of Heinrich Heine Alle (3 stops). You'll need to buy a ticket. Its a Euro-something each way.

If the weather is nice (i.e. sitting outside) and you like seafood, I've always had a good lunch at the Fisch Haus, on Bergerstrasse basically right at Flingerstrasse. You're close to the Rhein there so you walk along the Promenade .

The fancy shops are mostly on the Königsalle (the "Kö") which is five or six blocks back towards the train station. I'm not much of a shopper so I can't give advice there, but I have heard the Schadow Arkaden mentioned and it's in the area also.