Yes you need a strainer between the tanks and pump.
I may have mentioned it but I install quick disconnects on my pumps and my sprayer ends. This allows for switching any pump to any sprayer or one pump between more than one sprayer. It also allows for quick draining of a system or pump. It keeps them from sitting there holding water or spray mix to prevent rust and freeze damage. I also pull my pumps off and drop them into a 5 gallon bucket of old oil. Since doing that for about 2 or 3 seasons no more pump rebuilds.