I sort of skimmed through the replies, so I might be repeating what others have said. Since you already have your first post set, and I see that you have run the other two for one of your walls, the 345 rule will get you where you need to be fore the back wall. If you look on a framing square, it has this stamped into it, along with a few other handy rules. How far back will the wall be? 6 8 and 10 feet are what I use most of the time for smaller areas. If one of the walls is one of those length, then it become super easy to figure out where to set the post. I have a lot of measuring tapes, so I just set three of them out to get it all lined up. You can also mark a board at 6 feet and 8 feet, then use your tape for ten feet. Once you get your corner set, double check all your other posts to make sure you are parallel.
For bigger structures, measuring from opposite corners is faster and easier. I use two long hundred foot tapes for this. I set the ends of each into the ground with a screw driver and create a large X. When both distances are the same, you have a perfect square.
I never try to set my posts at the exact same height. It's very difficult and 9 times out of ten, you get it wrong anyway. I set all my posts, then use a straight board and a level to find the lowest one, or mark all of them at the same height. It's always 100% accurate doing it this way. Odds are good that if you try any other method, you will still end up cutting them in place to get them perfect, so why waste your time being frustrated?
Your beams across the front opening need to be sized according to the span, or distance they are covering. How far apart are your post? Anything over 4 feet and I'm using something bigger then 2x6's for my beams. A pair of 2x8's will probably be good from what I'm seeing in your picture. Nothing less will hold up over time. The same thing applies for your joists that will form your roof. How far are you going? If it's 12 feet or less, 2x6's 24 inches apart would be the minimum. And if it was me, I'd probably use 2x8's if it was more then ten feet, and go up to 2x10's if it was over 14 feet, 2x12's for 16 feet and trusses after that.
I've always found that the cost of the next size up lumber to be insignificant to the overall cost of the building. I've also done a lot of jobs that cost my clients a lot of money to fix what would have only cost a hundred bucks or less if they had used the larger sized lumber when they built the roof.