Trying to cut out a logo .png file that I need to print on shirts

A client wants me to print their logo on t-shirts. It was sent as a .png file.  I am using corel X4 and a graphtec Ce5000-60 with the 09 blue tip pen.  I want to convert the .png to a file where I can use it to print onto t-shirt vinyl to press.  I converted it to a JPEG but it came out jaggedy around the blending blue areas.  I need to fix this before printing.  I attached the logo for viewing.

Correctly, how would I go about getting this complete.  Would I create the file to bitmap, then break apart, then print each color separately, then piece them together when I am ready to press onto the t-shirts?

Other issue is when I trace the logo it has a blending in the blue circle area but it traces jaggedy and I did a sampe print and it came out jaggedy.  How can I get this smooth?  Then would I need two vinyls (blue) very close in color to show this slight blending in the logo?

New to the business.  All help appreciated.

Thanks