In addition to the violin, bow, and case, that typically come with a student violin rental, there are a few things you'll need to get started . . .
Required. You'll need rosin of some kind of rosin to be able to play. The bow hair just won't grip the strings without it. There are hundreds of fine rosins to choose from. But, for beginners, you may prefer something inexpensive with a sturdy case, like this RMD Student Light Rosin. (Rosin is extremely fragile.)
Semi-Required. Although some teachers and players prefer to play without a shoulder rest, most find that they need one to hold the violin in a natural way without hunching the shoulder or keeping unwanted tension in the shoulder and neck. Some beginners get by with an old kitchen sponge. But, we highly recommend investing in a proper shoulder rest. It will be much more comfortable for you, and will allow you to concentrate on learning, rather than holding, your instrument. Again, there are many different styles of violin shoulder rest, and you should consult with your teacher. But, for beginners, we recommend the Everest EZ rest, in the appropriate size. It sits firmly on the violin, without tipping over as many rests do, holds the instrument well, and is inexpensive.
Tuner or Pitch Pipe
Recommended. We recommend avoiding complicated tuners, as they can be more trouble than they're worth and confusing for young players. But, you'll need some sort of reference pitch for each of the four open strings, so that you can keep your violin in tune. Pitch pipes work well, though they tend to wear out quickly (which is why we don't carry them). A tuning fork is the usual solution for professionals, who first tune the A string and then the other strings to the A by ear. But, pitch forks are typically only available for the A, and the beginner will need the G, D, and E pitches as well.
Recommended. Even if you have a piano, we recommend a music stand for your young violinist. Most teachers ask students to stand while practicing, and you'll want the music to be high enough to read easily. A simple folding music stand is usually sufficient for beginners.
Cleaning / Polishing Cloth
Recommended. An old shirt works well for this - any clean cotton cloth. We don't recommend that students polish their instruments, so polish itself isn't necessary. But, it is a good idea to gently brush off the rosin from under the strings as a regular habit after playing, before putting the violin in the case, to prevent damage to the varnish. You might also consider the same for the shaft of the bow, though that's more for cosmetic purposes, as the bow isn't varnished and won't be damaged by rosin.