How do I care for a carbon steel knife?
Wipe your blade along the flats with a clean, damp towel (or sponge) - set upon the counter or cutting board - frequently when cutting acidic items like onions, tomatoes or fruit. Japanese sushi chefs will have a wet kitchen towel folded into a neat pad on a lower corner of their cutting board for this purpose. When you are done with one cutting chore and not yet ready to move on to the next or wash the knife, wipe with the damp towel, and then with a dry one similarly folded. Do not delay washing the knife once you're done using it. Wash and dry it before the meal, not after.
Work on your knife's patina. It's a layer of mottled gray oxidized metal that develops on your blade, but will protect the steel from full blown rust or pitting. Gray is good, rust is murder. You can force a patina (search online for "mustard patina" or "hot vinegar patina" for various schools of thought on the topic), or encourage it to develop naturally - the easiest way to do the latter is to scrub your blade with baking soda and a disposable kitchen scouring towel (like a Scotch Brite cloth) to remove stains after washing the knife - be sure to watch your fingers, and rinse and dry when done. After a few weeks or months of this, the metal will dull, and dark stains will come off in the wash rather than requiring to be scrubbed off. This is the patina at work. Japanese chefs will often scrub their knives to a mirror finish nightly with a scouring powder and rust-eraser rather than let a patina form - but this is a lot of work, and can damage the blade if done improperly. Many European and American chefs find patina appealing aesthetically - the mark of a quality knife used well.
Wash the knife with warm soapy water using a dish-mop or a sponge-on-a-stick now popular in the cleaning aisle of most grocery stores, and rinse briefly but completely. Do not use a dishwasher, or immerse the knife entire into the dishwater. Dry carefully with a towel - folding the towel on the counter, and pulling the flats and spine of the blade across to dry, and then move to the handle, holding the blade securely at the spine, before storing. 
If possible, store the knife away from sources of humidity: the sink, dishwasher and cooktop. A drawer block in a drawer with a few silica gel packets would be ideal. If the knife isn't used frequently, a very light oiling with mineral oil (available over the counter from any pharmacy) will keep it from rusting. Barely dampen a paper napkin with the oil, and brush lightly over the knife, wiping away any excess with a paper towel. Some wood handles (non-stabilized olivewood and walnut especially) are better off with occasional oiling as well.
Steel the blade frequently using a ceramic hone to take advantage of the edge-holding ability of carbon steel. Asian knives are typically sharpened briefly on very high-grit stones in lieu of steeling before use, due to asymmetrical or highly acute edge geometry - most cleavers, gyutos and santokus may be steeled on a ceramic hone, though. Check with the manufacturer or reseller.