The asked question:
The Sagulator is the generally accepted reference for determining if shelves are going to be strong enough to support the expected load over the desired span. Your table is nothing more than a very deep shelf, so lets take a look:
For your situation of
- a 29" clear span (36" overall minus 2 @ 3.5" 2x4),
- a 3/4" piece of Eastern White Pine (and option for "white wood" which is SPF - Spruce, Pine, Fir - i.e. whatever is cheapest on the day we made the purchase) on a 4.5" wide shelf (just one single piece of your top)
- 30 pounds of weight distributed evenly
The Sagulator shows a sag of just 0.01". That will hardly be noticeable. Increasing the shelf to to the full 18" depth of your table top (4.5" * 4 boards) yields 0.00" of sag.
Your desktop is sufficiently strong for a laptop, monitor, a pile of school books (or other reading material), and whatever other detritus of life collects upon it.
The unasked question:
A more pressing concern is racking - horizontal forces applied to the table that could cause it to come apart.
If this was simply nailed together and I needed to disassemble it, I'd flip it over and push the legs sideways until they came apart. Racking is the unintentional application of that process while it's standing upright. Assembly with screws will provide more resistance to being pulled out, but the wood holding them in, especially SPF, can still give way resulting in a collapsed table.
Making the assumption that this will be pushed against a wall, I'd add diagonal bracing on the wall side. Taking 2 2x4s, cut a 45°* on each end and put them at a diagonal from the bottom of each leg toward the center of the table top. You can put these on the side that will be against the wall - they do not need to be centered in the 18" depth of the table. Screw them in and you'll go from a bit wobbly to rock solid.
While you're at it, you may want to add diagonals within the 2x4 boxes that make up the legs. I'd put one in each leg, one running from the top rear to bottom front of one leg and one running from bottom rear to top front of the other leg.
Your comments also included questions about using metal bracing with diagonals. One of these diagonal braces that you've described should also serve the purpose of bracing against racking. You can use whichever looks better or, if it's to be in the middle of a room instead of against the wall, whichever provides more leg room.
Based on additional info in the comments:
Based on the need to pull this table over the arms of a chair, I'd suggest flat metal brackets like these.
Image provided by Lowes.com. No particular endorsement of this brand or store implied.
Apply one leg of the bracket to the 1-by and one leg to the 2-by supporting it (so they're on the "face" as you pull it toward the chair). Applied on each corner, these should provide sufficient racking resistance while allowing maximum clearance.
These particular brackets have 3.5" and 4" legs. I would think that would be enough to prevent racking. You may want to see if you can find ones with longer legs, especially since this table will be moved on a regular basis. Be aware that as the legs get longer, the width of each leg will increase to provide the necessary strength. Be sure to get ones that do not have legs that are wider than the 3/4" actual width of your 1-by material.
While you're at it, you could also use a set of 4 on each leg (joining horizontal 2x4 to vertical 2x4) to prevent them from racking as you drag the table toward you and push it away. This may actually be better than the diagonal 2x4 recommended above, since it will firmly resit racking in both directions for a table that will be moved regularly. The width of the legs for these won't matter much since you're very unlikely to find any brackets with legs wider than the 1.5" of your 2x4 material.
But these ain't pretty!
I recognize that these brackets are not particularly pretty in any way, shape, or form. You could hide them by routing/chiseling/sanding a recess to set them into, then covering the entire face with some edge-banding/veneer, but that may be beyond the need or your abilities for this particular project.
If you had not already glued your table top together, you may have been able to cut a recess on the inside of the boards and embed them between the two outer boards on each side, but A) it's too late, and B) that would have taken some sort of filler block on the inside of the leg for them to mount to.
* You don't indicate the height of your table, so the 45° angle is just a guess. Cut the angles as necessary to ensure that both braces will fit. If it's tall enough that a piece at a 45° doesn't reach to the center, it's still OK, that will provide enough diagonal bracing.
