There are two aspects to this. One, you need to make sure to have structured data applied for the schema.org classes that have Google rich results, these include articles, recipes, products and reviews. The full list of rich result eligible classes can be found here.
In order to be eligible for the rich results you need to satisfy the Google identified required and recommended properties for the specific class. Google has a beta rich results test, which you can use to verify your eligibility. Note it is only a beta feature and therefore does not currently search for all potential rich results.
On the other hand, Google determines when it will show rich results, unfortunately having the markup present does not guarantee they will appear. First, check to see if Google has crawled the site (which you can see in Google Search Console) and then be patient. It can take several weeks.