What type of wrench should I buy
A Stillson wrench that you referenced is normally called a "pipe wrench" and comes in sizes from 6" through 96". The most usable sizes for a home owner would probably be sizes from 8" through 18" inch. They are normally used to hold or turn smooth type piping. Due to their flexible, top jaw they do not work well on soft materials like plastic pipe and fittings. For the home owner, a pliers looking tool commonly referred to in the piping industry as "channellocks" (pardon the use of a proper name, but that is what they are called by tradesman) or the older name "water pump pliers", would be a great choice. You may also want to buy an adjustable wrench called a "crescent wrench" to hold the parts with the flat surfaces. By the way, when you replace the ball valve make sure you use a quality ball valve made in the USA since most of the foreign ones are of very low quality. I am not trying to belittle or disparage any manufacturer or supplier, that is just the way it is. Also that reducing fitting on the left side of the picture should be replaced if that is a domestic water line. Only brass, copper, or plastic fittings and pipe should be used on domestic water supply.