Nails plus glue into endgrain vs screws plus glue vs dado (for shelf)

To answer your question, without actual joinery like dovetails or dadoes, I'd recommend glue-and-screw. That is, some mechanical fastener that actually gives those butt-joints some strength. Because, while PVA glue is incredibly good, it is not at its best with all of the joint being 50% end-grain. Many joints like dove-tails, fingers, dadoes, or rabbets would offer some long-grain to long-grain surfaces to mate, which is where the best glue line will be found.

Similarly, grooves for the vertical pieces will aid in placement and glue-up, and will give you some strength against racking forces. But another benefit would be more long-grain as part of the glue-line, even though both axes would be less superior end-grain-to-long-grain.

The 1x4 or whatever in the back is there to guard against racking side-to-side, and would almost certainly need to be screwed unless you did a full or half-lap into the vertical pieces. You could get away with no mechanical fasteners if you had a nice clean half-lap, but I'd use at least two properly installed screws. This part would offer a less superior glue-joint, but its main job is to offer mechanical strength against racking, especially with a tight lap joint.

In short, if you don't want to make any joinery at all, you should use mechanical fasteners, and I'd recommend screws for simplicity and speed.

(An side on screws. Remember that it is the head of a correctly designed wood screw that gives you the strength in wood, so you want sufficient threads fully engaged into the second piece, with the top piece essentially only being clamped to the second; few threads should be actually engaged in the top piece. Think of the screw as a local clamp to ensure good glue joints.)

It also occurs to me that pine or other soft "white" wood will really want to cup and twist, and there is nothing in this construction that will dissuade this from happening. Since this is not heirloom furniture, that might be acceptable. But, accept that this may not sit perfectly flat, and may even split as the wood moves perpendicular to the grain as the seasons change and indoor spaces change in both humidity and temperature.

The alternative is to consider gluing up panels, alternating the grain to minimize cupping in the same direction. You can also buy softwood dimensioned panels so this is done for you already. I'd actually recommend buying these pre-made panels for this reason.