I generally use Taranaki Andesite as it is a hard stone and has enough variation to keep it exciting . All stone types have their unique qualities and possibilities. Very few stone types are totally uncarvable. The following are types of stone that I have used and some of their significant pros and cons. These are my opinions. Other artists may have different experiences.
Andesite - volcanic hard stone. Carved from boulders, each is different and can vary within. Colours range from pale grey to nearly black, including pale pinky brown to dark reddish brown. Heavy and durable, andesite will outlast my lifetime with little to no weathering. As a rule the harder, tighter grain (paler grey) will take a higher polish, while the more 'coloured' boulders are more porous and less likely to achieve a mirror finish. To maintain andesite, a quick waterblast or spray with outdoor anti-mould/fungal/lichen type product will suffice. Sealant may be used to retain a pieces wet look or to retain its lighter dry colour. Sealants will need to be redone over time. Andesite is HEAVY.
Matatoto Andesite - volcanic hardstone from a specific site - no longer available (aside from a few pieces stashed in various studios). A gorgeous soft sage green, with occasional quartz seams and darker streaks. Challenging to work, it does not achieve a noticeable mirror finish but can be brought to a lovely soft matte finish.
Hinuera - soft, sandstone. It can vary from light cream to buttery and may have variations within. Fun to work with but will weather over time. Edges will soften and fine details may be lost if unsealed. Sealant will need repeating as often as every 3 years. Unsealed hinuera outdoors will go black and quickly grow mould and lichen etc. Spraying will help but waterblasting is not recommended as it will remove the surface of the piece. Lighter than andesite. NB hinuera used in commercial applications has been baked and so will retain its colour, becomes harder and is more durable.
I have also carved a little granite, basalt and some marble. Each has its own characteristics.
WOOD I will try almost any wood. I prefer to make pieces which are for indoors as wood does not last long term outside without a lot of ongoing maintenance. Working with aged, dry wood is different from green timber but both have challenges. My current preference is to use wax based sealants as I prefer a soft, tactile finish which still feels wooden.