The links you're talking about are called sitelinks. Google's official page on the topic can be found here.
There is no way to guarantee that sitelinks will appear. Google decides whether or not they want to show them, based on how useful they think it'll be to the user, and how high-quality they think your site is.
You can increase the chances of getting sitelinks in your SERP listing by:
- Ensuring your site is well-structured. You should avoid having lots of very thin pages that have similar content to one another, for example.
- Carefully choosing the anchor text and
alt attributes of your internal links. You should have very clear navigation, and you should plan your internal linking structure carefully, so that it's clear which pages are the most important and useful.
As you can tell, this is quite a big topic. You probably want to read around the area of Information Architecture in SEO. Read about how internal links affect site structure and crawl issues, and then take another look at your own site to see how things could be improved.