Apart from those who buy their own, warbird pilots are either ex fighter pilots, or pilots with extensive aerobatics/display experience.
Knowing somebody involved in the warbird scene who can introduce and vouch for you obviously helps but, even then, only if you have military/aerobatics experience.
There's no shortage of pilots offering to fly warbirds but owners are understandably cautious given the safety considerations, the value of the aircraft and insurance considerations.
Realistically, working as a volunteer is unlikely to lead to flying anything more powerful than a Harvard - unless you fall into one of the categories listed. That said, volunteers who prove themselves genuinely committed over a period of time do get rides on positioning flights, with a few aeros en route!
For most of us, the Harvard was our introduction to warbirds but, even if you go no further, it's a great aircraft.
If you are serious about learning the fly a Harvard, learn to fly a tail-dragger properly first- a Chipmunk is ideal. I also got a few hours Stearman time which certainly helped with the feel of a big radial engine, but is not essential.
I'm sorry if this is discouraging, but owners' caution is sensible. My close friend, Mark Hanna, was killed in an Me109 a year ago. He genuinely was a natural, one of the best and most experienced warbird pilots in the world. Little did I think when we were building experience in a Pilatus P2, and then a Harvard, years ago that it would ultimately lead to such a tragic loss.
On a lighter note, if you get the chance to fly a Harvard for an hour - go for it. It will give you an exciting insight into warbird flying.