What is required to start sculpting clay (tools, experience...)?
There are a few different options for sculpting materials that fall withing the broad definition of 'clay' 

Water based clays

These are essentially the same stuff used for pottery and are widely used for 'fine art' type sculpture. The big advantage of water based clay is that its plasticity and hardness can be controlled according to how wet it is so you can use wet clay  to quickly rough out a form and then allow it to dry a bit to firm up to allow you to refine surface detail. 

Their disadvantage is that they tend not to be very string when wet and usually need some sort of internal support armature to support their own weight. You can get clays filled with paper or nylon fibres which are a bit stronger when dry but, in my experience aren't that nice to work with as the fibres can interfere with the surface. 

Oil based Clays 

Also known as 'Plasticine' common professional brands are Chavant and Monster Clay 

These are available in a range of hardness and, unlike some cheaper brands can be used with RVT silicone to make moulds. 

These tend to be a bit firmer than water based clay and as such lend themselves well to small, finely detailed models and is also a bit cleaner to work with. It has reasonable strength and tends to be less reliant on internal armatures. The downside of this is that it can be more time consuming to build up the bulk of the model and tends not to be so good for sweeping organic surfaces, although then can be softened a bit with heat and the professional brands can be melted and cast. 

These give very good detail and can be smoothed very effectively. They also don't dry out and as such can be worked into and reused more or less indefinitely (although they will gradually decline in quality over time).

Modelling Putty 

This includes epoxy putties like milliput these are mixed from 2 parts to a firm putty and will dry hard after a few hours. This has the advantage that they are hard and string enough to be used as a finished piece and a such are frequently used for converting and modifying existing miniatures. 

Obviously the downside of this is that you only have a few hours to sculpt it before it sets, although you can still drill, sand and carve it once it has set which can be useful in some situations. 

Wax 

There are a variety of sculpting waxes available. Their big advantage is that their harness can be controlled easily by both heating and blending different wax grades. They can be both sculpted and carved depending on relative hardness. 

Waxes especially harder ones tend to be good for getting very crisp detail. 

Tools 

There are a few broad categories of sculpting tools 


Wax carving/dental  tools : these tend to be stainless steel and are good for fine detail. I would say that a set of these is a good starting point. 
Ball stylus tools : designed for creating consistent fillets but can also be useful for sculpting fine detail and undercuts. 
Hardwood clay tools :  these tend to be geared more towards larger pieces in relatively soft media. 
Silicone tipped shapers : best suited to final smoothing and polishing, specifically in conjunction with solvents and/or lubricants. 


etc etc 

Tool selection is really a case of what works for you but all of the above are likely to be potentially useful.

Casting 

Casting is a whole separate subject in itself....

Silicone rubber is probably the most versatile mould material and is reasonably easy to use. 

For small scale casting you are really looking at catalysed or 2 part resins which include Polyester, Epoxy and Acrylic. In general PU resins are a good bet as they cast well, aren't too expensive, have good mechanical properties and are reasonably easy to use. 

Figurine casting also tends to use resin with powder fillers added. Different fillers give different properties to the casting. 

PVC and ABS need to be extruded or injection moulded and aren't really practical for small scale casting. 

General Advice

I would suggest starting out with a relatively soft medium such as water based clay or the softest grades of oil clay. These are easy to work with and will let you 'sketch' figures reasonably quickly to get a feel for working in 3 dimensions. As you progress you may want to try harder grades to allow you to get a smoother finish and crisper detail. As your style evolves you will start to learn exactly what properties you need.