The first step in this project is to install the posts, and that means digging holes. Make a hole at the center of the bed and one at each end of the bed. Bury 2'-3' of each post so they'll be really sturdy. Each of the holes has to be the same depth or the posts will be different heights. Use a measuring tape to get them even.
Once the holes are dug, place one post in each (Image 1). Use the post level to get them plumb. Once they're where you want them, start backfilling the holes with dirt. Tamp down the dirt with the handle of the shovel, making sure there are no air pockets. If the soil in your garden is so loose that the center post won't stand up, you can set the post in concrete: put the post in a bucket and then pour concrete around the post to make a moveable support (Image 2).
To support the side posts, attach a 2x4 to the bottom corner of the planting box using a 3" wood screw (Image 1). Then angle the board to the post to form a triangle. Attach the 2x4 to the post with three wood screws and then go back and add two more wood screws to the base. Add another support brace to the other end of the bed. A trellis full of foliage acts like a sail in a storm: make sure any trellis you build is supported to hold up under heavy winds and rain. Screw the 2x4s across the tops of 4x4 posts using the same 3" wood screws you used for the side supports (Image 2). Two or three screws per post are enough.
Once the boards are up, you're ready to install the trellis wires. Eyehooks are a fast and inexpensive way to secure the wires (Image 1). Mark where you want each wire to go using a laser level so that the hooks will line up. There'll be seven on each side, starting from the top. The hooks screw into the wood until all of the threads are buried and the neck of the hook is just outside the face of the wood. Once all of the eyehooks are in place, install the trellis support wires (Image 2), starting at the top and working your way down. Thread the wires through the first eye and tie off by wrapping the wire through the loop several times and then twisting it to secure it. Pull the wire to the corresponding hook on the opposite post and tie it off using the same loop and twist technique. Once all of the wires are in place between one side post and middle post, go the other side and repeat the steps.
Once the squash trellis is complete, build a second one for the gourds. Each trellis covers a whole bed. Once you've built one of these, it's easy to duplicate the process; you can you can use as many posts and wires as you need to fill the space. You could use redwood or cedar posts instead of pressure-treated pine; you could also substitute plastic mesh for the wire lines. Always choose materials you feel comfortable working with and types that you can find easily.