The centerfire suggestions are great, but I have to say "+1 to Buzzcook and Natman". I also suggest getting a .22 and training first. As I've posted elsewhere; my suggestion would be a Marlin 795. It is as accurate a you can get unless you spend at least twice as much. You can put a scope on the 795 or a set of peep sights (ex: Tech Sights) for less than $200. Then you head to a local Appleseed event
$125 (795) + 60 (Tech Sights) + 15 (sling) + 5 (elbow pads) + 20 (bulk box of ammo) + 20 (RWVA membership) + 80 (local Appleseed weekend) = $325. I don't believe there is a more inexpensive but effective introduction to riflery.
From there go with a centerfire. But meanwhile a person will have the opportunity to try some different rifles and find one that is both commonly suggested as being a 'winner' and
really trips your trigger.