[Water Warrior 2]

Any motorcycle specific 10-40 oil will do. You can also use a non moto specific oil if you want. You must however be sure they do not contain any ingredients such as friction molifiers and labelled as energy conserving. Dino or synthetic is your choice but I have had very good results with plain old Shell Rotella 15-40. It is cheap and certified for moto use by Shell. You could also use Rotella synthetic 5-40. Still cheap compared to other synthetics.
Filter wise you can go to Suzuki or elsewhere. HiFlo make the HF136 filter for the GZ and is a common filter for many smaller engines. Check out local small engine repair shops, auto parts stores or online for the best price and availability.
You will need 1300 mm of oil for the change. Oil and filter require 1400 mm of oil. Buy two qts and you have lots left over to top up with when needed.
Be sure to check the oil level in the sight glass with the bike held upright. If it is leaning on the side stand the level will be incorrect and you will overfill it.
Keep in mind that some oils will change color soon after an oil change. That not a bad sign. It just happens.