Case: gives you a place to store your ‘ukulele when you are not using it. It also protects it from dings, scratches, and the occasional drop. Hard cases are the best protection for your ‘ukulele. Soft cases are okay for protecting your ‘ukulele from scratches, but do not really protect them from being dropped or having things set on them. If you really value your ‘ukulele, a hard case is your best bet.
Tuner: is a way to make sure that your ‘ukulele is in tune not only with itself but with all other properly tuned instruments. By far the best kind of tuner is a clip-on kind. There are several brands you can buy. I use the Intellitouch PT1 Tuner It works great but is a little bulky for an ‘ukulele headstock. The Korg AW-1 is another; Herb Ohta Jr. uses this one. Stand alone tuners that pick up notes using a small mic are also a widely-used option. I don’t believe they work as well because they pick up noises other than your ‘ukulele, but they are great for beginners because they are cheap. You can also get tuners that plug straight into your signal path when you are plugged in.
Capo: clamps down on all of your ‘ukulele’s strings to create a barre. It is very handy when you are trying to play a song in F# (or some other weird key). All you do is put the capo on the first fret and play everything in F. I don’t know of anybody who manufactures ‘ukulele capos, but Kyser makes a banjo/mando capo that works very well.
Cables: are used for plugging your ‘ukulele into an amp. They come in many different lengths and two kinds of jacks. The longer the total length of cable you are running, the more signal loss you will experience, so try to keep it under 20-25 ft. The 1/4 in. phone jacks that are used for instrument cables come straight or angled. Try plugging both into your ‘ukulele and see which kind works best for you. Keep in mind that angled jacks are nice for plugging into effects when you don’t have a lot of room to work with.