For more formal competition, either select the sport/event/discipline you're interested in and get a rifle that fits the rules, or get a rifle and find a sport/event/discipline that it will fit into.
For example, 50-foot iron sight prone matches wouldn't suit a 10/22 without making an Anschutz inexpensive in comparison.
I think the best advice was to look at what the other competitors are using and pick something at least similar.
Old Mossberg M44/144s, H&R M12s, Remington 541Xs, and the CMP Kimbers would all be relatively inexpensive bolt gun choices, from ~$200 or so on up, and they'd have a bit of history too. Keep stock style in mind, too.