A standard double loop chain is often used for porch swings.
Assuming three hefty occupants (~250 lbs each) and a 100 lb swing, you're up to a load of 850 lbs. If you use 4 lengths of chain to connect the swing, that's 212.5 lbs per chain. So you need at least 2/0 chain, which is rated to a working load of 255 lbs and an UTS of 1020 lbs.
EDIT: If you're a nerd/engineer like me, you can get more technical and take into account angle of the chains from vertical and normal acceleration from swinging. Then your equation comes to:
4*T*cos(Θ)-m*g=m*a where a=v^2/(l*cos(Θ))
or
T=m*((v^2/(l*cos(Θ)))+g)/(4*cos(Θ))
where
T is the tension per chain length using a total of 4 chains,
m is the total mass of the system,
a is normal acceleration,
l is the chain length,
v is maximum horizontal velocity you'll be reaching, and
Θ is the angle of the chains from the vertical
which will be a bit more than 255 lbs if you're swinging very fast, or have very short chains or a very wide swing. In that case, you may want to go with 3/0 chain.
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User is right saying you should cut the rough shape with a band saw. If you don't have one, you can use a jig saw but use the most rigid blade that will follow the form. The next step will be to use a table mounted rotary sander/ drum sander to work the piece to the scribed line. It will be next to impossible to make a well shaped piece without this step. Check out a few episodes of "New Yankee Workshop" online (PBS) to see the techniques. Keeping a full length of grain top to bottom is extremely important,(as mentioned) otherwise the piece will split in time. Good luck with your project.
Best Answer
Virtually any 4x4 lumber will be more than strong enough. Pressure-treated SPF would be fine, as would untreated pine/fir or cedar. You won't break a good-quality 4x4, as each is only carrying roughly a quarter of the load.
Screws are almost unnecessary here, but almost anything that fits the holes well is fine. They carry very little load and simply prevent the thing from falling apart when being transported. 1/4" lag screws or heavy construction screws come to mind.