One of the easiest ways to ensure you get a good beginner instrument is to go through a reputable violin shop. Violin shops are specialized and employ luthiers - they really care about quality and really know their stuff!
I recommend Kennedy Violins and Shar Music often because they ship nationwide, have great quality at some of the lowest price points, and have great rental and rent-to-own programs.
I also highly recommend Benning Violins in Studio City - even just for a visit!
Violins come in fractional sizes for children, and getting an instrument that is the right size is essential to success for the student - it's nearly impossible to learn to play on an instrument that's too big!
Sizes are notated like fractions: 4/4 is a full size (for an adult, or older child or teen who's about 5 feet tall), and smaller sizes include 3/4, 1/2, 1/4, 1/8, 1/10, and 1/16.
Kennedy Violins has a helpful size chart. While we can guess the average size based on a child's age, measuring their left arm will give the most accurate sizing. Just remember, if they are between sizes, always, always go with the smaller size!
If you find a used violin that's well made, the right size, and in good condition - go for it! Violins actually sound better once they've been played for a while, so a new violin isn't necessarily better than a used violin.
However, there are now a lot of manufacturers out there that sell very cheaply made violins and there are also a lot of people selling these, too (whether they didn't end up using it or discovered that they don't work well). These can be difficult to spot since they look nice enough in pictures! But they are typically very difficult to tune, have a poor sound, misshapen fingerboards, poorly shaped bridges, unbalanced bows - they can make a beginner's job of learning to play very difficult!
I'm happy to answer questions about finding an instrument - or take a look at something you found, if you're just not quite sure!