Once a page's content or a site's primary domain name has been changed, you should wait for Googlebot to re-crawl and re-index the page.
This is the natural method for how most content is updated by Google. Sometimes it can take a fairly long time, depending on how frequently Googlebot currently crawls the page in question.
Once Google have re-crawled and re-indexed the page, the old content will usually not be visible as it will be replaced by the current content. You don't have to do anything specific for this to take place. It's generally not possible to speed up crawling and indexing, as these processes are fully automated and depend on many external factors.