Both have pros and cons. ....... Most like to stain, I don't. Stain is an easier prep, but unless you are disciplined to re-stain every two years. you will degrade your deck sooner. If your stain looks like it needs to be redone, it is already way past due, and moisture has been allowed to penetrate the wood. The film build of stain is just too thin for me. ...... Were I forced to use it, I would lay down 3 coats at the start just to equal one coat of paint.
Neither would I use a 'primer'. ........ Yes, I grew up with the 'oil primer' on wood story. Even used to tell it when in the business, before I decided it was marketing BS. ....... I have never used (which I have on my shelf) an oil base primer on wood in over 20 years. ....... Top shelf exterior house paint, done. (Satin minimum sheen)
My only exception would be IF you had 'ponding water' on wood. In that case I doubt I would even use primer, but instead do two coats with an exterior alkyd.