I sort my white and very light colors out and wash them together. I use OxyClean in addition to HE detergent and they stay white. I wash them in cold water most of the time because some of the clothes say use cold water and I discovered that warm water causes fuzz balls on some of those.
I also sort my clothes separating the more heavily soiled from the lightly soiled.
The reason hot water worked better for rinsing when SA Mamma H had to hand wash her clothes is that hot water does more easily rinse out the detergent/soap than cold water.
It"s important to not use too much detergent. My Dad, who both sold and serviced appliances, said that most women use too much detergent. If your water is not hard try using 1/2 the amount suggested on the detergent container. If your water is hard still use 1/2 the amount but add a powdered water softener.
I use the second rinse cycle on my machine just to be sure all the detergent is washed out.
Detergent left in the clothes does make them dingy. My Dad demonstrated by washing a clean load of whites using only water softener and the water became sudsy and after a second rinse the clothes were brighter.
I rarely, if ever, use bleach and I like my clothes to be bright. I wear most white and light colored clothes one time only so that there is very little dirt build up.
For the swim suit, wash it every time you wear it. When several of us go swimming I put all of them in the washer, on cold water, and run them thru the gentle cycle without using any detergent. I do sometimes put in OxyClean if they're looking dingy. Never dry them in the dryer. They lose elasticity but you probably know that.