Is a "painting pad / sponge square" indeed the way to get a smooth wood surface after applying polyurethane?
That sounds like water based polyurethane. Water based polyurethane is more difficult than oil based poly. You can recoat water-based poly with oil-based. 

If that sound like something you want to do, then the first thing to do is let your shelves cure for 2 months. Then use fine (150 then 220 grit) sand paper. Clean the shelves thoroughly; dust with a cloth dampened with mineral spirits.

Ventilate the area well and wear a respirator rated for organic fumes (follow the label warnings on the polyurethane can) when applying the finish.

Sponges and brushes are just different. Either one takes getting used to. Personally, I often use a lint free cloth over a sponge and I replace the cloth often. I am not saying that my way is any easier than another. Only practice makes perfect.