Really you should wash them the same way you would with cotton cleaning cloths. There are a number of ways to wash things and I think any reasonable one should be fine. Wash them separately from other items. If your rags get oily, go to the laundromat to clean them.
If you are worried about soap residue, use "safe" detergent like Laundry Soap Flakes, but it may not clean every type of grime you may end up with. Air dry.
Microfibre is a synthetic medium so bleach or alternately a Peroxide-based "Oxy" cleaner should be OK (may affect colour though).
I wash everything in cold water; solves a lot of laundry problems before they start, and I don't find I need the hot water to get laundry clean. Warm or hot water can also act as a fixative to set some stains (eg protein based ones).
I'm a lifelong Tide user, never used Gain which is a low cost detergent, but if it gets your clothes clean then maybe I'm the one wasting money. Having said that, I am personally wary that "value" detergents would work in cold water as well as Tide.
I am a reluctasnt convert to microfibre; I've started using them but only on items where I am not concerned about scratches, no matter how tiny. So, not on camera lenses, not on paint jobs I care about; for that I still use 100% cotton (I buy them from the diaper service after 50 baby bottoms have worn them to super softness). But all the above regarding washing applies to the cotton rags as well.