Protect Yourself/Your Family From The Scourge Of Plastic
First I should say, investigate, be suspicious. If clothing has some claim to fame, such as that it is wrinkle free (embalming fluid) or moisture wicking (https://www.shape.com/fitness/clothes/harmful-chemicals-hidden-your-workout-clothes), you can be almost certain it's not good for you ! Unfortunately this applies to a lot of products sold nowadays, not just plastics. Other than that, here are some ideas:
 
1. Avoid buying plastic or using it in all its myriad forms. This is very hard to do as many natural products have pretty much vanished. Some are available on the internet.

2. Do not buy synthetic clothing (polyester, nylon, rayon, acrylic, fleece etc. even spandex which replaced elastic along the way. Spandex is almost impossible to avoid.) The clothing industry is beginning to come up with alternatives (such as lyocell, tencel, modal, etc.).  These are made with fibers produced from such things as tree trunks/bamboo soaked in chemicals - however they do not leach plastic fibers. Do NOT use synthetic blankets and sheets, especially fleece !   All night long, as you turn in bed, you will be releasing and breathing in lots of fibers, especially from the blankets.

3. Do not recycle synthetic clothing. Until such time as there is a hazardous waste program in place, put it in the garbage. Clothing made from recycled fibers degrades much faster than the original product. I put mine in a plastic bag which I tie tightly to slow it's entry into the planet.

4. Cover heavily trafficked areas of synthetic carpet with a natural fiber rug (wool, cotton, jute). Also cover the areas where children play. Replace the carpet with something better as soon as you can (tile, natural carpet). Hold your breath when emptying vacuum dust  or dryer dust into the garbage !

5. Keep your food and drinks covered if they are sitting out. There are waxed cloths you can buy or use a clean cotton dish towel, even aluminum foil.

6. I segregate my clothing, I make a cut in the label of the synthetics, where possible so I can discriminate. I wash synthetic on the delicate cycle and then hang to dry. I do not put them in the dryer. There are filters you can buy to trap the fibers in the wash but that residue will go in the trash and end up leaching out, albeit more slowly. 1,900 miniscule fibers leach from each synthetic garment, at a minimum EVERY SINGLE TIME IT'S WASHED. Some thick fleece produce 2 million at a time.

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Plastic was developed in 1907 but did not come into large scale production until the 1940's. In the 1970's the first synthetic clothing was produced and since it gives the textile industry more profit, it is now used in billions of garments. Plastic production has exploded with approximately 150 million tons at the present time ! So in a relatively short period of time it has polluted the entire planet and everything on it, even the bodies of insects. It's psychologically difficult to protect yourself against microscopic objects especially when the danger comes from everyday objects but the fibers are now in the water (rain/ocean and fresh), in the air you breathe, both indoors and out, and in your food.

The garment industry has been aware of the problem of synthetic microfibers for almost 20 years. They have immense lobbying power and so far have managed to block even feeble attempts by lawmakers to bring it to the attention of the public.