I’m trying to create a shortcut for Ancestry website.
I want to open multiple urls contained in their
pages.
I used
Get contents url
Get url from input
Open url
When it produced weird results I added quick looks.
It seems that even though I’m running from a logged in screen, shortcut first get contents is handed their login screen. Whereupon it then opens all those urls 
If you are getting content using Shortcuts, the browser aspect of Shortcuts would need to be logged in to the Ancestry web site. It sounds like you are logged in within the Safari app, but for security reasons, the logged in session details are not shared outside of Safari with other apps, and vice versa.
I.e. Being logged into a web site in Safari does not share that login with Shortcuts, where you are a user who is not logged in.
As a non-logged in user, when you then try and access the URLs which require an authenticated user, you get redirected to the login page.
Hopefully, that explains your observations.
OK cheers.
So how do I log in using shortcuts
That depends on the login method used by the web site you are accessing. It may be your only option is to pop-open the login page in a Show Web Page action, but it could well be that if you then close it to progress to the next step in the shortcut that it loses the login details.
Some web sites allow for OAuth and there are roll-your own solutions for that if you search this forum.
Some web sites offer an API to get data, and this is the ideal approach for automation as the whole thing is intended to be used for hands-off automated access. Accessing web pages with web page logins is not designed for automation and is one of the reasons you can run into challenges like this. If it is ancestry.com that is the site you are trying to access, then it would be mixed news on this front. They do have an API, but they apparently don’t make it accessible publicly, only to partners.
I understand that you want to solve this using Shortcuts, but want to note that Shortcuts is not the best tool for the job (since Ancestry has no public API, as pointed out by @sylumer).
Two tools better suited for the job are Keyboard Maestro and Selenium. Both are mac based though, so they might not be of use for you.
Edit:
What information do you seek from each website? One semi-manual solution could be to: