Selecting a Violin
First, a quick note about size: did you know that violins come in a variety of sizes? Adults play on full size violins, but growing children need smaller, fractional instruments to prevent pain and possible injury from playing. Your private lesson teacher or a violin shop employee can properly size your child in person. or you can look up sizing guides and charts online.
For beginners, especially growing children, I recommend renting a violin outfit (violin, bow, case, and accessories) from a local shop. In addition to being budget-friendly, getting started on a high quality, properly set up, and properly sized instrument is beneficial to the beginner student. The first few months of learning to play are challenging enough - why add to the challenge by using a cheap, poorly crafted instrument? Here in the Twin Ports, Christian Eggert Violins and Schmitt Music both have good rental programs. These shops are also able to do repairs and provide maintenance services for your instrument.
For more experienced players, or if you'd rather buy and own an instrument from the start, I recommend getting the best violin (and bow!) that you can afford. Generally speaking, a higher price tag does mean better quality. Expect to spend a minimum of $400 on a decent violin and $100 on a good bow; higher quality violins and bows range into the thousands. This is not to say that you need to spend thousands for a violin to sound good... in fact, there are plenty of very nice sounding violins in the $500-$800 range. But if you can afford it, investing more is well worth it! With violins, you do get what you pay for, especially in the $400-$1400 price range. Push for the best that you can afford and keep in mind that violins will retain their value as long as they are played and maintained. Do plenty of research and try playing many different violins at several shops before making a purchase.
Instruments acquired through other means (gift from a friend or family member, thrift store find, purchased through Amazon or any other retailer that doesn't specialize in string instruments, etc.) will likely need to be brought to a luthier for proper setup and/or repairs. While I am able to diagnose the issues that violins and bows commonly have, I do not have the skills or tools to address repairs like bow re-hairs, bridge re-shaping, and seam glueing, all of which are best left to the experts! Depending on the extent of what your instrument needs, repairs and setup could be relatively fast and inexpensive, or prove to be more costly than the violin is worth. If you didn't get your violin from a shop that specializes in string instruments, or if it hasn't been played in years, you may want to have it checked out by a luthier (Christian Eggert Violins and Schmitt Music are both good options) before your first lesson to ensure it is in good working order.