Hello Huntinhick. Welcome!
I have Handi's in several calibers and am still dialing some of them in. So far the .223 is averaging about an inch, sometimes less at 100 yds with the Winchester White box 45 gr. commonly found at Wal Mart, Bass Pro, etc. My .44 Mag is punching one hole groups with as many shots as I want at 50 yds. One of the locations I hunt is very thick and a 50 yd shot would be about the longest I could get. The .30-06 is a work in progress. Its best groups are 1"-1 1/2" at 100 yds. But that's not every time. Most of my groups are three shots.
The recoil on each is quite managable. The .44 is about that of a 12 ga., but a bit sharper. Not a push. The .30-06 a bit more, but not abusive. Now, my 12. ga Tracker II. Holy Cow! Without a doubt the most brutal firearm I have ever fired. Nuf said.
I agree that the .223 would be a good first choice. Ammo is cheap (NEF does not recommend using military 5.56mm), recoil is very light and you can find them just about anywhere, new or used, which eliminates having to order one.
Good Luck!
EVOC ONE