1) Keep the cornett away from direct heat and strong light - do not leave it either in full sun or by a radiator. Even temperature and humidity are ideal conditions for it. After a playing session, dry the instrument out with a pull-through (a silk oboe pull-through is ideal), and if you have been playing for a long time it is best if you leave it outside its case for a few hours to dry at a natural rate.
2) The wood has been treated to provide a certain degree of seal and to make it fairly water-resistant. Cautious blowing-in of new instruments will, however, help to promote a longer life: suitable blowing times might be about 30 minutes per day for a week, 45 per day for the second week, and then fairly freely thereafter. If after a couple of years you feel that the wood (especially around the fingerholes) is looking very dry or 'starved', an oiling with a fine oil like almond oil would be a good idea. To do this, pour oil down the instrument, making sure it gets into the fingerholes; leave it sitting for about half an hour; then clean it all out. IT IS NOT A GOOD THING TO LEAVE OIL SITTING IN A LAYER in the instrument; it will get sticky and eventually crystallise, collecting any particles that are around.
Leather-covered instruments: the leather should be rubbed down occasionally with a soft cloth. If it seems to be becoming very dry or cloudy-looking, a good quality black shoe polish can be applied, and rubbed well in. If the instrument gets accidentally caught at the end and a little leather peels off, use a small amount of ordinary white PVA glue to stick it down again.