Welcome to HAZ AZKendog. I think you'll find more information than you could possibly hope for on this site. Keep asking the questions - and all of us crusty hikers will try and give you the skinny on what is going on.
Well, as far as day hikes go - around the Payson area there are a bunch of dandies. Horton Springs (a trail that has generated a lot of postings recently) springs (sorry
) to mind - and anything along Tonto Creek is always a good bet. Check out the TrailDex for some of the ratings and distance.
As for good places to see wildlife - well, you're going to just have to trust your luck on that one. I've traveled all over, and I've seen wildlife in some unexpected areas, but I've also been in areas that I thought sure were prime for wildlife, but got nothing.
The best bet is to hit more out of the way trails - places that don't see a lot of traffic. But then, if you're new to hiking - I don't recommend that approach at first. Get a few miles under your belt on some more well traveled trails first. That will help you build some trail finding skills, and also get you to begin to find your own comfort levels and physical limits.
After a little while, when you've built up the hiking skills, give some of the more out of the way spots a shot. It is well worth your time.
'Weird is a relative, not an absolute.' - A. Einstein