If there are no water weeds, you will need a net. Even a home-made plankton net can sample small animals at or near the surface of the water.
The main parts are a towing line to pull the net along, a swivel at the net end of the line to stop the line kinking as the net spins while it is being pulled, a stiff hoop to hold the net open, a very fine net (think about what you have in the scraps basket at home), three lines attaching the hoop to the swivel and a small glass bottle which attaches to the lower end of the net, a metre or so beyond the towing ring.
You can buy a swivel from any fishing equipment shop. The hoop can be a length of coat hanger wire, bent in a circle, the mesh can be the footless leg of a stocking, either stitched or stapled over the hoop, or even glued to the hoop with contact adhesive (wear gloves and work outdoors to avoid the fumes if you use this glue). A stapled net won't last as long, but it's easier to make.
As you pull the net along, any animals trapped in the open mouth will be pushed down to the end, where they will be largely protected from damage by the still water in the glass bottle. You can buy nets like this for a high price, or you can make your own very cheaply.
The towing line can be a fishing line: I have used a fishing rod to haul a net like this through the water while walking along a wharf or jetty (the rod stops the net from snagging on the pilings), or you can tow one from a boat which is being pulled along by a 2 hp motor.
If possible, fix the bottle to the net by attaching a metal or plastic screw lid at the narrow end of the net, so you can change bottles regularly, just by unscrewing them. That lid will need a hole in it, but if you use a standard jar, you can have plenty of spare lids. Otherwise, tie string around the stocking and the bottle, and pull it very tight.
With a suitable net, you can explore the plankton types and densities over a time period: either looking for daily patterns, or monthly patterns (some plankton may respond to the full moon, so samples taken regularly at 9 pm could be useful). Maybe there are patterns you can see as the seasons change.
Nets like this can also be hauled through seaweed and water weed to sample the small animals living on those plants. This is likely to damage the net, so use a replaceable but strong one.
You could also just explore the types of plankton found in one place, or compare different environments at more or less the same time of day, over a period of time, to see whether any observed differences continue over long periods. Aside from that, you have the tool, you have some ways of using it, so go for it, remembering that the most interesting questions are always your own questions!
The origins of the towing net
Nobody knows now who was the first to develop this handy item. John Macgillivray, writing in the 1850s, thought it worth explaining how one was made, so maybe the idea was new back then:
Not having seen a description of this useful instrument, I may mention that the kind used by Mr. Huxley and myself, consisted of a bag of bunting (used for flags) two feet deep, the mouth of which is sewn round a wooden hoop fourteen inches in diameter; three pieces of cord, a foot and a half long, are secured to the hoop at equal intervals and have their ends tied together. When in use the net is towed astern, clear of the ship's wake, by a stout cord secured to one of the quarter-boats or held in the hand. The scope of line required is regulated by the speed of the vessel at the time, and the amount of strain caused by the partially submerged net.
—John Macgillivray, Narrative of the Voyage of H. M. S. Rattlesnake, vol. 1, chapter 1.Or maybe it was only new to Macgillivray. As early as 1768, Joseph Banks makes mention of using both a "cast net", and when that was lost overboard, he attached a hoop net to a fishing rod. Perhaps this was just dipped into the water, but the idea of towing a net seems obvious enough. Like a lot of simple ideas, most people probably thought it not worth mentioning or explaining!