l was thinking of using a synthetic oil

l do not know what a Suzuki 250cc gsxr is or looks like, but the best oil would be 10-40, it has really good high temp protection, and good low temp starting performance (viscosity).

l do not think that bike qualifies as a crotch rocket unless that is a 2-stroke engine in the chassis? You could use synthetic oil, though its totally unnecessary unless you drive a lot of miles each year. Just make sure to get a good brand name oil. l am using Mobil Clean 5000 10W-40 in my 2006 GSXR 750, it is supposed to be good for 5000 miles. l will never ride 5000 miles in a year and synthetic is usually good for 10,000, but you should change the oil annually at least.