The order I would recommend for initial firearms purchasing would go something like this:

(1) .22 rifle

(2a) .38/.357 revolver (midsize to large, not small)
-or-
(2b) full size 9mm semi-auto pistol

(3a) mid power semi-auto rifle (7.62x39 or 5.56x45/.223 or 5.45x39)
-or-
(3b) 12 gauge pump shotgun

(4) .22 pistol or revolver

(5a) full power bolt action rifle (.270, .30-06, 7.62x54R, .308, etc.)
-or-
(5b) .177 caliber air rifle

(6+) where ever your heart takes you - larger bore pistols, small concealed carry handguns, additional rifles, additional action types (lever, single shot, etc.) ...