Rather than what 3 calibres, its more of what three rifles? And slightly depends on where you are and aspirations. I think you need:
1) small game, vermin, practice and general shooting fun rifle. Needs to be cheap to run. 22lr fits the bill in spades, but could also include 22 hornet, 223, 17 hmr etc. Sporter weight barrel and scoped to allow you to do all you want.
2) a reach out and touch at longer range type rifle - may be for vermin, deer and smaller game, but may also do smaller mountain stuff as well as targets etc. Probably heavier barreled, scoped for longer range, easy to shoot prone etc. Depending on desired usage could be any calibre from 223, through 243, 25-06, 6.5cm, 270.
3) Your do anything, go anywhere rifle. This is sporting rifle in the traditional sense. It should be well balanced so you can take a fast shot at a running pig, or snap shot in woodland, yet able to take a steady shot at 200 - 250. It needs to be light enough to carry all day long, yet heavy and balanced well to make shooting easy. You should be able to shoot a small deer without mincing it, take a red stag on the open hill, but shoot a big moose, eland, lowland red stag when you get that call from an old friend -“ somebody has dropped out get here tomorrow”.
It will be well handling rifle, will have open sights, and a do everything scope. Or perhaps a do everything and a specialist woodland / driven type scope. Calibre 7x57, 7x64, 30-06, 308 etc. All can shoot a light bullet fast and flat, but equally a long heavy for calibre bullet for bigger stuff and thats where the 7x64 and 30-06 come into their own. 6.5’s could be included as well as 9.3’s and 375’s and depending on where you are and with modern bullets and loads either could work.