How to add a "read more" link on User profile teaser display?
This you can do using Views and providing the code for Link on User page, follow following steps.

1. Creating the Complete User profile at default Drupal account page:

1.1 Add the all the fields on the Default user-profile what you want on fullDisplay(Admin/Configuration/Account setting/Manage field). So any user when creates a account, he will have all section to fill-out. For example My Full user-profile has the following fields.


Name
Picture
Bio


2. Create the teaserDisplay

2.1 Hide the fields which you do not want on the teaserDisplay. So, on the User-profile setting page(Admin/Configuration/Account setting/Manage Display), now hide what you do not want to show on Teaser Display. Now for my setting I want to Hide Bio. So I will just hide this field at Manage Display.

So current user-profile field settings are:


Name
Picture
Bio - Hidden


Now, till this stage, we set our Drupal's default account page as teaserDisplay but simultaneously we created those fields which we want on fullDisplay, hence we kept hidden but we will display all those hidden fields using Views.

3. Create the Views-fullDisplay

3.1 Create a new view page, Select USER under Views setting. Under pager, choose 1 item.
3.2 Now on the setting page, add all following setting as shown in teh following image. So here we are accessing that Bio field which we hide on User profile page.


  



Contextual filter setting: Under contextual setting select User: User ID and under that select Display all results for the specified field.
Path: Now as shown in the view set the path /user/%/view-page. I have kept last part here, /view-page as I was just trying previously, you can change to full-profile, but you have to change that at all places.


4. Setting the Link on the teaserDisplay page (so we can re-directed to fullDisplay)

To provide the link, the link should take the User-Id from the URL and can able to pass that User-Id in views URL, so Views contextual filter can provide the result of that User, from where we have redirected. Hence, to solve this I raise separate question and you can follow answer as shown below. I am not providing answer as you have to just copy from there.

@ShawnConn has provided answer for how to add link on user-page. Full Credit goes to @ShawnConn for this answer.

How to get current path for user profile field?

5. Output Image:

teaserDisplay- Default User Page


  


Full Display - Views with bio field


  


Now, this way you can add as many as field you want, just hide them so they does not display on teaser display and show in views so they display on fullDisplay page.