Pound stakes into the ground at the four corners of an 8-foot square and tie builder's string between the four stakes. The stakes represent the patio cover posts' locations; adjust the size of your structure as desired, but note that larger post spacing might require larger support beams and joists. Consult your building authority to determine allowable beam and joist spans.
Measure the distance between the square's diagonal corners. If the diagonal distances are equal, your square's sides are equal and meet at right angles. Adjust the stakes' positions until the diagonals are equal. Dig post holes at least 12 inches deep and 2 inches larger than the diameter of your tube forms at each stake. Footing depth varies according to frost conditions; consult your building authority for site-specific guidelines. Fill the bottom of each hole with approximately 2 inches of gravel.
Set a tube form in each hole and use a level to adjust the horizontal position of the forms' top edges. Forms should protrude 2 or more inches above grade or as specified by your local building authority. Cut tubes with a utility knife if necessary. Pound stakes outside each form and run string between the stakes. Attach a line level to the string. Check if the forms are level with one another and adjust their height.
Backfill the sides of the form holes with gravel. Fill the forms with premixed concrete and level the concrete's top surface with a trowel. Plunge a post base connector into the center of each form, aligning and leveling the connectors with a string line. Allow the concrete to cure at least 24 hours before attaching posts to the post base connectors.
Set the 4-by-4 posts in the post base brackets and use a level to plumb their vertical position. Attach the posts to the brackets with a power drill, driver bit and the bracket manufacturer's approved fasteners. Pre-drill bolt holes through the support beams and the tops of the posts. Drill two holes per post. Each set of two beams will span between and sandwich two posts. The sets will run parallel to one another on opposite sides of the patio cover's square footprint.
Hoist the support beams to the posts' tops. Align the beams' ends with the posts' outside faces and the beams' holes with the posts' holes. Slip a washer over each bolt and slide the bolts through the holes. Slide another washer onto each bolt and then a nut. Tighten the nuts with wrenches to secure the beams to the posts.
Lay out the positions of the 2-by-4 joists at 12-inch intervals, perpendicular to the support beams, beginning at the ends of the beams. Fasten the joists to the support beams with the nail gun, using at least one toenail on each joist's face. Lay out the position of the 2-by-2 slats, perpendicular to the joists, on the joists' top edges. Position the slats at 1 1/2-inch intervals. Fasten the slats to the top edges of the joists by nailing through the slats' top edges with the nail gun.