Keep the keys clean by wiping them with a soft cloth after playing, especially if your hands have been hot and sweaty, and always play with clean hands!
The areas around the keywork and rods will get a bit dusty after a while - the best way to tackle these bits is to use a smallish artists paintbrush, which can get into the nooks and crannies.
For the wood I use a special wax which I got along with some other things from America - you just use a little dab and rub it in, then buff it with a cloth. If you ask at your music shop you can probably get something similar. Don't use furniture polish though!
You'll need to oil the bore occasionally (but not for a while if it's a new clarinet). When it starts to look dull inside, rather than smooth and shiny, it probably needs oiling - but do be sparing! Again, you should buy special bore oil rather than use any old thing.
If you do a Google-search for the Clarinet BBoard, you can then click on the 'Sponsors' list there to look at the 'Doctor's Products'. That's where I got my things from, along with other goodies like mouthpiece bags and his own brand of silk swab. He also does Gonzalez reeds which are very good.
I've been really pleased with the things I've bought from him.
The Clarinet BBoard is also a really useful resource - I've learnt an awful lot about technique and about clarinets in general from there.