Yarn & Hook
There are a whole host of crochet specific yarns out there, but you can crochet with any yarn and even with plastic bags, twine, wire or fabric!
When you're starting out, I recommend a DK (double knit or worsted in American terminology) weight cotton yarn in a light colour. This will be smooth to work with and will create loops and stitches that you will be able to see clearly. Something like King Cole Bamboo Cotton DK which comes in a great range of colours, has a smooth surface and isn't prone to splitting. For a slightly finer weight yarn (4ply or sportweight), try the DMC Natura Cotton range.
Once you've selected your yarn, you'll need a hook. Hooks range in size from 0.5mm to 25mm and beyond. They come in a variety of materials from plastic, to steel, to bamboo and rose wood. A good place to start that will keep your initial outlay to a minimum while you're learning is the Pony Crochet Hook range; but if you want to treat yourself, then the colourful wooden Knit Pro Crochet Hooks are a delight to work with.
For DK (worsted) yarn, I would tend to go for a 3.5mm hook. As you gain in experience, you can try different hook sizes to change the tension of your work. You can also check the band from the ball of yarn that you are planning on using. Often, it will show the recommended knitting needle sizes only, but as a general rule, the yarn will crochet up well in a hook size which is 0.5mm smaller than the recommended knitting needles. By way of example, the band on the King Cole Bamboo Cotton DKrecommends using 4mm knitting needles. So if we subtract 0.5mm from 4mm, we get to the 3.5mm crochet hook.