The first instrument is often a music keyboard. These are reasonably priced and stimulate an interest in music even in very young children. Ask your school music teacher for advice, because some of the very low cost keyboards are very limited. The higher priced keyboards play notes softer if they are hit less hard, more like a piano.
Your child may enjoy the sound of a particluar instrument. This can be a major motivational factor, so it is a good idea to go with the child's ideas, at least initially. Many children change the instrument they play in the first few years of their music education.
Many children may find it easier to play woodwind instruments like a clarinet than brass instruments like trumpets and trombones. Woodwind instruments do not require the lip vibrations that brass players need.
Most children's choice of instrument is a matter of chance. It depends on what instrument is favored by visiting music tutors.
There are more professional violinists than players of any other instrument. This is one reason parents often encourage their children to take up the violin. Buying a violin is a tricky business.
You can buy a violin for less than 70 dollars on the Internet. Should you buy it? Never.
A violin is not just a pair of shoes. Shoes are designed to be used and thrown away when they wears out. A well-made violin will last for hundreds of years. The 70 dollar violin above should just be thrown away. Buying one for your child is a waste of 70 dollars.
If your child is taking up the violin, or any instrument, it might be an idea to rent rather than buy, until you see how your child takes to the instrument. That is the point at which you should consider buying.
Buy the best instrument you can afford, usually here at musician's friend. Price is definitely linked with tonal quality. Listen to a range of violins, pianos or trumpets being played by the musician in the retailers. You will notice differences between the different instruments. Always buy a musical instrument you and your child like the sound from. Musical training will attune the child's ear to any deficiency in the instrument you buy, and this will detract from the pleasure he or she gains from playing it.