Before I get into sizing and types of guitars for kids, please know that although I do not rent guitars, I do have a ½-size guitar that kids can use during a trial lesson. I also have an extra full size guitar if you need to use that. Then, if you decide guitar lessons are a good fit, you can purchase your own. But what to buy?
Ah, product research. Sometimes it's fun. Like when you find you suddenly need a turbo-charged convertible (or so I'm imagining). But children's guitars?...maybe not as much. Fortunately, I have done a little research for you to get you started.
First off, know that guitars come in in ½, ¾, and full sizes. Most adults need full sizes. For kids, it is especially important to have an instrument of the correct size. Unfortunately, kids often play guitars that are too big for their bodies. Learning guitar is hard enough in its own right, so make sure your child has a guitar they can comfortably manage. The chance of success and enjoyment is greatly increased.
The following recommendations are rough guidelines based on age. A child's torso height and arm length are the real determining factors, so use your best judgment and "try on" a guitar before you buy it if possible. Usually, kids 10 and under will benefit from a ½ or ¾-size guitar. Typically ½-size guitars work well with kids 3 to 5 and ¾-size guitars with kids ages 5 to 10. However, to further add to the confusion, ¾-size guitars often come in two different lengths, 34" (ages 5-8) and 36" (ages 7-10). I should also note that another option for 3 to 5-year-olds is a baritone ukelele. They are smaller than a ½-size guitar and only have four strings, which are tuned the same as the first four strings (the higher ones) of a guitar.
Once you have determined the proper guitar size for your child, you need to think about quality. Most guitars for kids often lack it, especially in the ½-size. One problem is that a poorly built guitar has trouble maintaining a consistent tuning throughout it's entire range. It's possible to find an acoustic guitar at a big box toy store for $20 -30, but the craftsmanship is suspect at best. I bought one of these alleged guitars for my kids and it turns out that it wasn't a very good idea. I would have been better off flushing the money down the toilet. I'd still be out the money, but I'd have one less piece of junk cluttering up my house.
I visited a couple of locally-based music stores near my neighborhood in Plymouth, MN and each had options for your child that you might want to look into. Of course, there are many more options online and in your own neighborhood. Also, please keep in mind that I do not endorse any of these establishments or products and I do not receive any kind of compensation.