Automated iOS and Android acceptance testing with RSpec and Appium can be set up by first installing Appium's dependencies and building the sample apps. Once installed, users can run example tests using `rspec simple_test.rb`. The test uses Selenium WebDriver client libraries to interact with Appium, allowing for automation of mobile app interactions on iOS and Android devices. To start a new test session, users need to initialize the Selenium client, find elements on the page, perform actions, and switch between alerts. The process involves setting up a "describe" block, initializing the Selenium client in a "before" block, quitting the Appium session after each test in an "after" block, and writing the core of the test within an "it" block. By following these steps, users can automate their mobile app tests using RSpec and Appium.