Hi
I stayed in Germany for half a year almost 8 years ago, near to Duesseldorf. I went travelling to a different city/town in every single weekend.
If you're travelling over the weekend, try the Schoenes Wochenende Ticket (literally means Happy Weekend Ticket). It gives you unlimited travel on RE and local city trains for a single weekend day at 37 euros for up to 5 persons. It used to cost only 40 Marks (around 32SGD) before the Euro. Only downside is, you can't take the fast Inter-City or Inter-City-Express trains. Otherwise, your train tickets to further locations can cost a bomb.
http://www.toytowngermany.com/wiki/Schönes_Wochenende_Ticket
For train timings, check their webbie
http://www.bahn.de
For places to visit, it depends on your interests..! Off my mind, some very scenic places I can remember... I'll include some pictures, sorry last time use P&S and no digital camera, so pics might be koyak. I'm a fan of castles.. heh
1) Koeln (for architecture)
Cologne in English. About 1 hour from Duesseldorf. Huge black gothic cathedral right above the train station. Thats the only thing of interest there, you can climb up all the way too. Bring wide angle to capture the whole structure and more.
2) Mosel Valley (scenic)
Quite a long distance from Duesseldorf... but one of my fav places, will go again. Its a very long valley of vineyards with many little towns. Check out Cochem Castle, Eltz Castle, or rent a bike and zoom down the valley. Check the direction, don't blur blur try to cycle up slope! Get a room at a guest house, cycle more or take a boat back to return your bike.
Cochem Castle
Eltz Castle
3) Fairy Tale Road (interesting)
Its a long road of little town where its said that the Brothers Grimm travelled and wrote their stories, such as Pied Piper, Hansel and Gretel, etc.
4) Koenigswinter (castle only)
Dragon castle on a hill.
5) Kassel
Hercules Temple
Enjoy your stay!!