There are four main types of ukulele: Soprano, Concert, Tenor, and Baritone.
The soprano is the smallest and is often the one most beginners choose. For adults I think this is a poor choice. The concert and tenor offer far more space for the pupil to play chords and as such I would recommend them.
The baritone is tuned completely differently from the soprano, concert, and tenor.
Here’s a video that shows the main ukuleles…
There are many different types of wood used to make ukuleles and the sound of the instrument is influenced by which type of wood is used. In terms of price I would recommend spending at least £50. Yes, you can buy them cheaper but they are often cheap for a reason!