(Boston, MA, USA)
How can I do this on copper, bronze or other metals best suited for this effect?
Rainbow effects can be painted on with a torch causing a heat patina or with different patina agents/chemicals such as liver-of-sulfur. Both techniques require practice and experimentation to achieve the look you desire. LOS rainbow effects depend upon the metal used, temperature of solution, metal and metal prep. A hot solution for copper and cold for silver. (a baking soda bath may stop the LOS process, but will not prevent the metal from oxidizing, thus changing your patina)
Most surface patinas will continue to oxidize and you'll loose those pretty rainbow colors. Some heat patinas are permanent on copper, others disappear over time unless you seal the metal/patina to prevent it from continuing to oxidize. LOS is not very effective on copper alloys such as brass and bronze.
Metal patination is a workshop in itself. Perhaps you can find a good book or tutorials on the various processes you wish to learn. Robin Cornelius has a couple tutorials for flame-painting copper and for rainbow patina on silver. Her Etsy shop is called, littlerock.