You don't need much to get started!

Yarn

Crochet hook. 

A pair of scissors. 

Tapestry needle. 

That's it!

Here's a little more detail.

Yarn

I recommend starting with inexpensive yarn. You don't merino wool or anything like that - just grab some acrylic yarn that feels nice and is in a color you like. Stay away from anything super skinny or super bulky for your first softie - and definitely stay away from any super shaggy or hairy yarns. They make it hard to see your stitches. 

Crochet Hook

I recommend getting a few different sizes. One of the "tricks" to making nice amigurumi is getting a tight enough stitch so that the stuffing doesn't show through. For me that often means stitching with a hook that's one size smaller than what the yarn wrapper recommends. You'll want to do a little trial and error to find out what works best for you.

This set contains three hooks - a G, H and J (that's 4 mm, 5 mm, and 6 mm) is a good set to start with. Those sizes will work well for just about any medium weight yarn.

Converting between US and other countries' hook sizes can be confusing. Here's a table that might help you if you're outside the US.

Scissors

You don't need anything fancy - just small sharp scissors. One of the things I especially love about amigurumi is that you can take your project anywhere. I love these small scissors. They have grown-up size finger holes, but they're small enough to fit in an Altoids tin, where I keep them with a few stitch markers.

Stitch Markers

These are SO helpful for keeping track of where you are in a pattern! You can't use the ring markers that people use for knitting - they need to be little padlock-ones that can be opened and closed.

I like these. They're a good all-purpose size for working with medium weight yarn. If you get into using really fine or really bulky yarns you may want to get some smaller and bigger stitch markers, but these are all you need to get started.

Tapestry Needle

Sometimes you need to thread your yarn onto a needle to attach pieces, and to weave in your ends. You want a thick needle with a blunt tip and an eye big enough to accommodate your yarn. This is the size I use and recommend.

That's really all you need to get started!