Hard question! AD 250 is during a period in which the Roman Army was undergoing various structural changes in response to the manifold challenges of the Crisis of the Third Century. That is, the Roman Army was transitioning in this period from the Army of the Principate (which I know a bit about) to the Late Roman Army (which I don't know as much about).
While a sagittarius under the Principate would probably be a member of a cohort of archers (sagittariorum) attached to a legion, from the early 3rd century it was possible he could be a specialist in a larger unit based on nationality, who could be used independently of the legions (Le Bohec 28). Additionally, archers were often recruited in units known as numeri that were not formally incorporated into the structure of the Imperial Army (97).
Le Bohec mentions that easterners (eg. Palmyrans) were prized for their skills in archery, which is fitting with the general reputation of the Persians for their skill in archery. So our hypothetical archer c. AD 250, especially if he served in the east, was probably a Syrian or Armenian or Palmyran (etc!) archer.*
Like the Cretan archers of the Greek and Roman Republican world, our archers was already a specialist in the use of the bow and would have had his own bow. However, the Roman Army had a policy of subsidising the equipment requirements of its troops and as it paid such numeri, I'm sure they had no trouble getting the state to pay for a new bow if they needed one. In case you're wondering, they used powerful composite bows.
Horse archers, which the Romans also increasingly employed in the 3rd century would have used a smaller, shorter ranged bow, were a different thing again, with a different skill-set, but as you might have picked up would also have generally come from the east, where such tactics originated. A horse archer would also not require the army to teach him to fire a bow, but would have learned it from a young age, or what use would he be competing against nomadic horse archers in battle?
*Indeed, under Severus Alexander (reign 222-235) the use of archers, especially from the east, apparently became more prominent (197). Apparently eastern archers were so valued that archer units were not replaced by locals, like other legionary and auxiliary units, but by more easterners (Southern 211).
Sources:
Le Bohec, The Imperial Roman Army, trans. Raphael Bate, 1989/1994.
Southern, The Roman Army: A Social & Institutional History, 2006.
Am I more likely to come from a certain social class or ethnic group? Do I get training in melee combat?what are the terms of my enlistment and are they different from that of a legionairre?