Love some classic varmint rifles.For what it's worth I would start with a .223 bolt rifle. Can shoot for hours and won't beat you up, as I aged I no longer enjoy recoil so spend most of my time messing around with the 22 varmint calibers.
I tend to buy used, many fine accurate varmint rifles have been made over the years and can be found for reasonable prices. Just one example is the Winchester Model 70 Heavy Varmint, made in the 1990's with a Wilson barrel and HS Precision stock, they have a reputation for accuracy, they were available in 5 calibers, mine is .223.
You can spend a lot of quality time at the 100 yard range tailoring loads to shoot small groups and if the opportunity presents itself can easily take a .223 out to 600 yards with the right loads.
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Don't count 223 out past 600 though. Did a prs tactical division custom rig for a buddy using a 7twist shilen and a long throat reamer print. 77 rdf noslers are in the 2.4" area seated to lands. Wicked shooter. It sees 1000yd airtime regularly. Spotting hits can be challenging but it's good training. Thinking about doing another one for myself.