Start moving up the row. There's a method to the madness of going from bottom to top--after the bottom panel is attached, you can use it as a guide by resting the next one on top of it and then pressing down on the double-stick tape. Afer you've finished one vertical row, go to the next row and start at the bottom again, and use your laser level to make sure you're not veering off at an angle. You'll get into a good rhythm here, especially if you have someone help you peel the double-stick tape. You'll find that the prep work for this wall is the most time-consuming part of it; putting it up takes just minutes.
As you're applying the wood to the wall, don't worry if the panels aren't completely flush. If your wall is a dark color you won't even see the gaps. Another thing you can do is to overlap the panels to get rid of the gaps. The wood is thin enough so that you'll hardly notice the overlap.
Nothing maintains this wall treatment better than a tack cloth. Keep your cloth in the plastic wrapper so it remains sticky, and periodically wipe the wood with it. For tougher stains, a damp cloth will clean the wood without harming it since you've sealed the wood with polyurethane.