May I suggest H&R/New England Fireamrs which have a combo rifle/shotgun.
They are single shot, break action weapons.
They are inexpensive and you may add sifferent barrels in different calibers with the center fire combos. Now owned and made by Marlin, they are a good quality, low cost option.
Rossi offers a similar package, the the H&R/NEF have better sights IMO.
Just an option if you do not plan on being the field very often for hunting
For the shotgun, I’ve used the Remington 870 for close to 40 years. It’s an extremely reliable design that’s been around for many years, plus it’s a very simple design and quite easy to maintain.
For the .22, I’d go with the AR-15 for a few reason. First it will satisfy your need for a varmint rifle and it is also satisfactory handling two legged varmints as well.