As with all small instruments, there are only two places a violin should be; in your hand or in its case. NEVER put an instrument or bow on a chair or on the floor - even for a moment - it's far too easy to forget it, or for someone else to come along and not see it. (A crushed violin makes an extremely difficult 3-D jigsaw!).
After playing, wipe off as much rosin dust as possible from the body of the violin, strings and the wood of the bow using a soft cloth. If the neck of the instrument has become sticky, clean with a damp cloth or changes of position can become very unpredictable.
Regular cleaning in this way will remove most, but not all of the sticky rosin deposits, and eventually slightly more vigorous methods will need to be employed. Experts seem to disagree about what is best to clean an instrument with. Proprietary violin cleaners are available, and my own personal experience has been that if used only occasionally (once or twice a year perhaps) using small amounts and little pressure in accordance with the instructions, no harm will come to the varnish. If you have a valuable instrument and have any doubts at all - take it to an expert.
Protect your instrument from extremes of temperature and humidity. The most common causes of over-heating are leaving the instrument in a car during the Summer (even in the boot it will become too hot), and storing it too close to a radiator in the house. Humidity can cause serious problems, particularly if the air is very dry as this can lead to the wood cracking as it shrinks. In England, this is usually not too big a problem unless working for long periods in recording studios, but in other parts of the world try to keep the humidity around 55-65%. Humidifiers are available to help with this problem.
Rattles and buzzes are very annoying and can sometimes be tricky to find. So that you don't have to bow the instrument and locate the cause of the problem at the same time, try tapping the tail piece lightly with your finger as this usually causes most buzzes to reveal themselves.
First, remove the shoulder rest to eliminate one possible source, then look for loose seams around the body of the instrument. Next, check the mute (if fitted), fine tuners and string protectors on the bridge. Less obvious is a problem with the chin rest - if it is too close to the tail piece, it will buzz against it (possibly only when the instrument is in the playing position and pressure is applied with the chin). Also check under the tail piece. Here the end of the tail gut may be a little too long and touching the body of the violin, or a fine tuner may be at the end of its thread forcing the moving part onto the body.
Always slacken your bow before putting it away. If you don't then the hairs will eventually stretch and you will not be able to tension the bow enough. Even worse, it can cause the stick to warp and twist. Take care not to drop the bow on its tip - this is close to where the stick is at its thinnest and may cause it to split or may dislodge one of the wedges holding the bundle of hairs in place at both ends.
Never touch the hairs with the fingers. No matter how clean your hands are, the natural oils from your skin will not only discolour the hair, but can also make it so greasy that fresh rosin will not stick to it resulting in a "blank" area on the bow. Bow hair can be washed in luke-warm soapy water, and although I have attempted it very occasionally, it is not something to be undertaken lightly. If you're not sure - don't try it!
More detailed information can be found in a useful booklet by Jane Dorner called "The Strad A-Z of Instrument Care for Players of the Violin Family" (Orpheus Publications). Details of this and her other books can be found here.