Here is a good video about squirrel hunting explained by some Louisiana girl that lives on the principles of living off the land:
Aside from her advice I have to add in my two cents about squirrel hunting having done it when I was 5-10 years old and just getting back into it last year. When I was young I used to live in New England and be ruthless with squirrels. I would chase them whenever I saw them and shake them out of trees because I hated them destroying my grandfathers crops. I had a slingshot with the wrist guard that created some pretty nice FPS on my shots. Those little buggers are much tougher than you think. I shook one out of a tree one time that was near the road and it landed on the asphalt, broke it's jaw and kept running. I've also tried to kill them with my slingshot several times. First off, squirrels in suburban areas will let you get fairly close to them. I've taken shots at squirrels from 10-15 feet away. Even so, they will most likely not die if you hit them. I have hit a squirrel with a nice size pebble before and all it did was hurt it. It did not break the skin, it did not bleed, it just kept moving.
You really need a BB gun or something with adequate FPS to kill a squirrel. As recommended in the video, if you suck at shooting a shotgun is your best bet. Shotshells with number 8 shot or so should do the trick, something light with a lot of spread with medium or no choke. I myself have a very tight bullet grouping on my .22 LR and am confident enough with my shots that I use the .22 because if you do use a shotgun you will probably get BBs in your meat on occasion when you're eating them.
I now live in Mississippi and I hunt the backwoods of the public wildlife management areas for squirrel. I successfully harvested my first squirrel last year and I was lucky enough to find one out in the forest. The problem is finding them. I have read that the best time to squirrel hunt is in the morning right after sunrise (when the animals want to forage for food) and I believe this is a good rule of thumb. Also, people say if it is raining hard squirrels will most likely fast and stay in their nests until it is clear weather, so I have also head it is best to catch them right after a rain when they are really hungry, looking for food frantically and disoriented.
I went squirrel hunting last weekend and I must have hiked 5 or 6 miles in between sitting, eating, and walking through the forest. The one squirrel I did see was too close to the trail that cars come in on to shoot. I did come 10 feet away from a skunk that I shot at. I believe I maimed the skunk, but I wasn't about to follow him and get sprayed finding out. Skunk are nuisance animals in Mississippi so I kill nuisance animals whenever I see them. How did I get that close? Standing still. People make way too much noise out in the forest and if you just sit still you will hear and see so much more than if you are moving around PERIOD. These are my experiences as a novice hunter.
IF YOU DONT TAKE AWAY ONE THING FROM MY POST TODAY PLEASE TAKE AWAY THIS POINT. If you really want to find animals in the forest do some scouting about their habitat. All animals need to eat and all animals need water (and yes some get their water from seeds, but lets not go there). From what I saw out in the forest area that I was hiking around in last Saturday, I know there are only a few acorn trees in the spot where I was scouting. As the video I linked says, sitting under a food source tree (like an acorn tree) is a great way to find critters. Even the skunk I shot at was only in the spot I was in because I was standing in a wild berry patch. My plan next weekend will be to bring a folding lawn chair, sit out under the acorn tree I found, and just wait for 4 or 5 hours. If there is one thing I've learned form sitting in tree stands, it is that you will see so much more activity in the forest if you just sit and be patient.
P.S. Yes trapping is more adequate, but that requires a separate license as well. I could go into trapping all day from studying up over at the trapperman forums, but yes if you want to harvest squirrels for food trapping is the way to go. I like to hunt squirrels for hunting experience so that I can figure out how to land big game.