I would like to use UAG to publish Sharepoint 2013. However, the behavior I wanted are:

1. Users externally will access the portal via example.com

2. UAG login page will show, and users will authenticate

3. After authentication, the page should proceed to the SharePoint application published.

I am not sure if this is possible given that I can only use 1 domain name. I am also not so sure what to do. After creating my Trunk, automatically the portal is created. How do I add the sharepoint app in the same trunk that will give users the experience I mentioned above?

I am open for any suggestion. I'm new to this technology, hopefully someone could help.

Very doable but there are a few steps to get this working correctly both on the trunk/application and Sharepoint. A good book for this would be "Microsoft Forefront UAG Mobile Configuration Starter" from PACKT, just bear in mind it was written for mobile phone but it has all the steps you require.