The first gun I ever bought was when I was in 6th grade. Walked or rode my bicycle 4 1/2 miles each way (Through tidal waves, volcanic eruptions, etc lol) to pick apples so I could buy a shotgun. I ended up with a Mossberg bolt action 20 gauge. From sparrows to deer, I shot 'em all. I'll start with assuming it is full choke and capable of 3" magnum loadings. For all game, use high brass loadings. Where pellet count is important, where one might use #6 pellets in a 12 gauge 2 3/4" round, the 20 gauge should be using #7 1/2 or go to the 3" loading to increase the number of pellets. One advantage I have found of the 20 gauge over the 12 gauge, is given equal shot loadings, the 20 will group slightly tighter, slightly.
If I were out shooting 'coons out of trees, I would start using turkey loads with hard shot and work down from there. The turkey rounds of today offer advance shot cups, some going 30-40 yards before separating from the pellets. Some also have fillers to aid in getting as much shot out of the barrel without deformation. Turkey loads will probably be 3" loadings and available in #4, 5, & 6 shot. I won't say one brand or another will work better from your gun, but my ol' Betsy thrives on Federal high brass.