A good way to get babies and young children interested in playing music is to buy them a set of drums or percussive instruments and encourage them to play along when you're listening to music. You could buy instruments that have been specifically designed for your kid's age group or get some small bongos, a set of maracas or a triangle to start them off with. The important thing is to get them experimenting with different rhythms.
The recorder, a tubular flute of the woodwind family, is a popular choice for children in schools across the Western world and has been played in one form or another since medieval times. The recorder is a good way for children to come to grips with playing a few basic melodies and familiarizing themselves with reading music. It's relatively inexpensive, has closely spaced finger holes so children won't struggle to play notes and works well as a solo instrument or when being played in a group.
A child can start to learn how to play the guitar as soon as she can get her hand round the fret board. It's best to start a child off with an acoustic guitar, if only from a cost perspective. You'll need to make sure that you're buying a model that's been specifically designed for your child's age and size. If the guitar is too big for your kid to play, she may lose interest and not return to it.
A child doesn't need an acoustic upright or grand piano to start experimenting with playing a keyboard-based instrument. It's easy to pick up a relatively cheap electronic synthesizer or USB keyboard that kids can play with to see if they have an interest or flare for the instrument. If your child takes to the keyboard, you can consider signing them up for piano lessons and upgrading the instrument as they improve.