I am building a attached lean-to carport. It is 21ft. Wide x 24 ft deep.
It is attached to the house. The span, being 21ft, needs support and I do not wish to use a center post in front, only the center and rear.
What size header beam should I use across the 21ft span in front with a 18" overhang?

You will have a hard time figuring how to run a 24 ft. roof joist without support. This is an engineering project with the support process you wish to use, not a common DIY project.

And will this have to deal with snow loading in climates where it snows, is it subject to high winds, etc.

And attaching it to the house may mean a lot higher increase of your property tax bill than a free standing one.
There's no way you are going to hide this from the town, so you might as well talk to the building inspector first about what you can do about the span under your local code, and then the zoning dept, and finally the assessor, to see what it will add to your tax.
(I think you will find that a front center post is much, much easier, and if you have room for 24' wide instead of 21', do that.)