Number one, I do NOT recommend you purchase an instrument online. I can tell you too many horror stories about students who have purchased instruments online that were unplayable. Besides supporting a local business, a music store can do a "setup" of the instrument and be there if any issues arise.
Size: This is a personal preference. I like concert size as a starting point for younger kids, but the instrument size you choose does not have much to do with YOUR size. Find one that feels comfortable in your hands. (Another reason to go to a local store!)
This video explains the different sizes that are available for ukuleles. (Please note - the video says that they recommend Soprano size for children - I disagree with this. Many adults play soprano!)
Model Recommendations & Where to Purchase
The following brands and models are ones that I have personal experience with, but are by no means the only decent instruments.
Don’t forget a case! A decent padded gig bag will cost around $20, and you definitely want to make this investment, since you will be bringing your own instrument to class!
Caramel Ukuleles Cost: Depends on the instrument! This recommendation comes with a caveat - you are ordering directly from the company in China, and I have had one instrument that I had to return because of defects. However, the other instruments I've purchased from them have been well worth the money, which has been FAR lower than most retailers. If you have the time to wait for the shipping, and your budget is such that you have some tolerance for the possibility of quality control issues, I'd consider one of these: caramelukulele.com
For example, their zebrawood concert at the time I'm writing this is $37.00. I own the tenor and baritone versions of this, and have been happy with them: Zebrawood Concert.
I have a ukulele at home, but it’s missing strings, or the strings are old? What then?
The sound of a ukulele is heavily influenced by the strings. In my experience, far more than guitars! Many low-cost ukuleles come "stock" with black nylon (look like plastic) strings. The sound of your instrument will be DRASTICALLY improved by replacing those strings with a good set of strings!
The strings I recommend are Aquila Nylgut strings (Make sure you get the correct size for your instrument!). They are available at most music stores for around $8.
I have also grown to love, and put on all of my personal instruments, a different type of string called flourocarbon. The brand that seems to be most available is the Martin M600 strings, and are only about $6.
I've included a short video below to demonstrate the difference in sound between strings, by taking a $50 ukulele and replacing the black strings that were on it when purchased to Aquila Nylgut.