We can favorite a question that interests us. Is there a way to favorite a user too?
In order to follow the posts of the user in question.
You can only subscribe to their RSS feed.
The link can be found at the bottom right under their activity tab. The RSS link is called "user feed".
Feature requests about following users, favoring users, adding them to a friend list or anything socially similar are and will probably always be declined.
Dan's answer is technically correct in every way but I want to remind you why there's no direct way to "follow" somebody on the site. When you follow people instead of topics, your votes start to become targeted. It gives your friends an unfair advantage over others just because of who they are, not on the merits of their posts' quality.
I've covered this in more detail before: Guidelines when voting for friends
If you are going to follow somebody via their RSS, I would suggest you either prevent yourself from voting, or at least make extra-sure that your votes are warranted, and that you're using the rest of the site normally too. If your account registers a lot of votes for one account, it will raise alarms and may even elicit the fabled rusty spoon.
Quoting from this answer on Meta StackOverflow
If you have a friends list don't you think you tend to upvote your friends because they are your friends, instead of the content of their posts?