I'm mechanic by trade, far from a carpenter and could use a little advice.  I'm looking to build a screened in porch (lean to style).  I'm looking to use the diagram below as a starting point.

Questions:

They used a 4x6 for the front support, wouldn't I be better off using a few 2x6's nailed together (in terms of warping?)  My front support will span 20' and I'd like to support it with three 6x6.

Also, how do I calculate what my new roof pitch will be?  I'm going to run 14' rafters from the ledger which is at a height of 10' out roughly 13' to the front support which I'd like to have max height of 7'-8'.

You will need 2x6 rafters on 2' centers, and you can use 2x6's nailed together for the beam. Just get 20 footers so you don't have to worry about splices. The 6x6's will be fine. Mount the base  of the posts in post brackets o they don't touch concrete. As for pitch, you really need at least 4inches of rise for 12 inches of run, anything less will require you to use roll roofing glued down.2 feet of drop in 13 feet is basically 2/12. Awfully flat for Ill.

If it's attached to the house you will need post footings dug to below frost (4 foot here in MN). You also will need to be sure your posts/beam/ joists meet the snow load requirements for your area. I would also check to see if you need a permit. Most inspectors don't fall for the "I didn't know" excuse anymore since the info is probably on your local city/county website.

Your diagram shows a 2x4 as your nailer (ledger board). Do not use a  2x4 go larger 2x8.
Now the diagram is not saying this but just by eye it looks to be depicting 2x4's as the rafters. Again, do not use 2x4's go larger. You might get away with 2x6's but given the snow load and load calculations for dimensional lumber based on 16OC it calls for 2x8's.

Are you going with pressure treated posts? I would not worry about warping, using a 6x6's is fine but you can get away with 4x4's.

As far as the roof pitch goes you generally want at least a 4/12 pitch to shed water properly but you are not going to get it with these dimensions.
Now I am not an expert but I think your pitch is going to be something like 2/12. As I understand it you would be better served with a metal roof for this pitch.  

The real question is do you need a permit and will there be a building inspector involved. If so, you need to stick to code to pass.

PS to add. Have you given any thought about how you want it to look. Is it going to be just a functional screen porch or do you want it to be aesthetically pleasing.  Are you going to have exposed rafters. If so, you might consider some nicer looking wood,a 2x8 looks like crap unless you paint it.  

Low slope like that will need mod bit.  Friends don't let friends use roll roofing.

Metal roofing might be an interesting alternative if you can increase the pitch.  Get that snow to slide off.


Consider ice and water shield running up under the siding?

I went all steel on mine. Drill stem pipe, 4" c purlins, R panel sheet metal.

Most of it is bolted together, but there are a few welds to tie it together.

If you want to stay wood, I'd at least sub in 4" sq tubing for the posts.