Honestly if this is to be your first gun , or if you have little experience with firearms then a .22 will be your best choice.
Shotgun for an animal in a trap is just a bad idea.
With a .22 , and I will add to that a BOLT action .22 you can develop proper shooting skills ... a ruger10/22 is fine but is a semi auto and the urge to pull the trigger several times sure is hard to overcome!
.22 ammo is cheap so you can shoot alot .... more shooting is a great way to get good
.22 ammo has no recoil per say and you will not develop the "flinch"
A bolt action rifle(or lever action) is 1. more accurate by design than a semi-auto . and 2. slows down the process of firring allowing the shooter to develop a better form.
A .22 is not a good choice for deer or hogs ... but for most all your problems a .22 is fine a 40g piece of lead in an animals head will pretty much stop its existence.
I've hunted and shot competitively for 25+ years alot of high power and long range experience , tactical matches .... etc. .. and I still find the .22 the gun we most shoot around the home ... its fun and cheap.
As for a brand ... A Savage MK2 bolt action is an inexpensive and SUPER accurate firearm ... they make one called a MK2 TR-SR that is short and quick to bring up , is light , and is accurate.... The stocks on the savages leave alot to be desired but as a work gun they are fine.