1. Hooks

First and foremost, you'll need to get some good hooks.  I've used a variety, from Susan Bates to Loops & Threads hooks.  My favorite by far would have to be Clover Amour hooks!  I've really taken a huge liking to their easy grip handles and hook shape.  They are very comfortable to use and I have not had any aches or pains in my hands from using these...contrary to my experience with some other brands!

Most - if not all - crocheters own a set of crochet hooks in various sizes, so I'd highly recommend that for beginners as opposed to buying single sizes individually.  One reason for that is because different patterns/projects call for different hook sizes!  For amigurumi, you might find that you tend to gravitate towards a certain hook size with a certain type of yarn, but then switch to another hook size when using a different weight yarn.  This is because thinner/lighter yarns tend to call for smaller hooks, whereas bulkier/heavier yarns tend to call for larger hooks.  If you're a beginner, you'd also be looking to find out what your gauge and tension is, meaning how tightly or loosely you naturally crochet, which will help you pick a hook size.  As an example, someone who crochets very tightly might need a larger hook size, and someone who crochets very loosely will need to go down one or more hook sizes to make sure the amigurumi project does not have any gaps or holes between the stitches.  (Gaps and holes would result in stuffing poking out or being visible in amigurumi, which is not a good look!)

2. Yarns

Standard yarns to use for amigurumi patterns would be medium weight (4), or worsted weight, yarns.  I also see DK or Light Worsted weight yarn on occasion, which is a light weight (3) yarn.  Information about yarn weights are usually clearly listed on the yarn packaging or in the listing description if you're purchasing online.  For reference, yarn weights generally range from 0 to 7, and below is a very nice table from Moogly to illustrate this!
For beginners making amigurumi, I would recommend using 100% acrylic yarns, as they are considered to be on the cheap end of the spectrum and usually come in a good variety of colors!  More expensive yarns, such as silk, wool, or cotton-blends, would be good options to consider for projects that will end up being worn (e.g., scarves, hats, clothing, etc).  I have only ever made amigurumi with 100% acrylic yarns that are specifically soft to the touch, and feel that they hold up quite well and you really get the biggest bang for your buck this way.

Below are my favorite yarn brands and types to use!

• Caron Simply Soft
• Red Heart Soft
• Vanna's Choice
• Lion Brand Heartland 

3. Stuffing

If you're making dolls, stuffed animals, and other toys, you'll likely need some fiberfill stuffing!  Sometimes, crocheters will add yarn scraps, or scrap pieces of fabric inside of their amigurumi as well. 

4. Yarn Needles

Most amigurumi projects require sewing pieces together, so you will also need some yarn needles.  They are large-eyed needles with dull points, and suitable for sewing with yarns.

5. Plastic Safety Eyes

If you'd like to give your amigurumi creations some plastic safety eyes, rather than sewing eyes using yarn, you can purchase a set of plastic safety eyes for a pretty low cost!  I like sets because I never know which eye size and shape is best until I'm ready to try them on the doll!  Smaller amigurumi projects tend to use smaller eyes, and larger ones, larger eyes, so if you're looking to make a variety of creations, a set of varied sizes would be a good way to go!

Other Important & Useful Supplies:

A mandatory supply that you'll also need would be a pair of scissors.  You'll need scissors to cut yarn when you're changing yarn colors, fastening off, or after sewing parts together.