In terms of materials, the most important things are good quality clay, a few sculpting tools and circle template for making joints. Other stuff will help you along the way but it's not essential as much as the three items I mentioned before. So if you're on a budget, invest only in clay. Other items can be as cheap as you can get! I'll share links to my favourite materials, but you don't have to get the same things. 

Clay - In this tutorial, I'm using "La Doll" air-dry clay. My other choices would be "Premier" or "Creative Paperclay".

Sculpting tools - I like to use an assortment of metal sculpting tools because you can use them for carving as well.

Circle template - very useful for shaping balls for joints and drawing joints in the blueprint.

Florist desert foam - to make a base that we'll remove later and turn the doll hollow. 

Sanding paper - it's good to have a variety of sanding papers from more coarse ones to very fine ones. Some of them will be useful to sand rough areas and remove imperfections, others will be great for polishing at the last stages of making your doll.

Watercolour brushes - you'll need just a few. A small round brush for sculpting and a flat one for smoothing the surface. I have a few techniques where sculpting with a brush is way easier than with a sculpting tool!

X-acto knife - for cutting and carving the clay.

Dremel tool - not an essential but makes the doll making a lot easier. If you plan to create more ball-jointed dolls I suggest investing in one. If it's just one-time sculpting curiosity you'll do just fine with hand tools. There will be a few sections at the very end where you'll need to drill holes. But the clay is quite soft so you can drill and carve it successfully using just the attachments of the drill. 

PVA glue - we'll be cutting some pieces apart and then glueing them back together. Simple PVA white glue works really great with air-dry clays.

Paper (masking) tape/washi tape - there are a few steps where you might need masking tape, but it's not essential.

Rubber bands - it's quite useful to have some on hand for holding the BJD pieces together.

Grey primer - I'm using Mr. Surfacer 1200 spray primer. But I've had good results with grey primer used for cars as well! This primer evens out small imperfections and prepares for casting the doll. If you won't be casting your BJD and will leave it as is, you can skip this step and paint it with white acrylics or white spray paint instead.