Latticework lends itself to a variety of decorative effects. To interplay round and square openings, bore holes at the points where strips intersect.
Vertical and horizontal strips make a strong grid pattern. If you don't mind a rough texture and variations in thickness, buy "fall-down" lath, inexpensive wood left over when lumber is milled.
Notched lattice strips create an interesting design motif. You can purchase panels made from notched lattice, buy notched strips, or notch the strips yourself.
Garden-spaced latticework doesn't provide a lot of privacy until vines begin to flourish. If you choose latticework for a trellis or arbor overhead, construct it with 1 x 2s, not lath.
If you don't like the looks of metal rail connectors, attach rails to posts with one of the joints shown here. All work equally well. Choose based on your tools and skills. To make a dado joint, cut away part of the post so the rail will be flush (or nearly flush) with the post. For a block joint, nail a short piece of 2 x 2 to the post, rest the rail on top of it, and toenail through the block into the post. For a butt and toenail joint, drive nails at an angle through the rail into the post.