I am wondering how to clean rings that have gemstones like rubies and diamonds, besides the jewelry cleaner.?

Like the jewelry cleaner you get from a jewelry store. Is there a good home remedy?

Dont put them in an ultrasonic machine ! Never with gems except very high clarity diamonds.
So, mild soap and a toothbrush are the best. Don't use high acidity soaps or any bleaching product since that might attack your metal settings and ring.
For daily cleaning I use a towel made from microfibers, like the one's for cleaning your glasses, to take the grease from the gems. (Diamonds love grease).

Enjoy daily your jewelry !

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