I do manual regression testing where I work and we use Testrail. When I first started working here, they had test cases in TFS(Azure) and I had to export them and import them into testrail (not a fun process). Our Scrum Team QAs/functional testers use Azure for their test cases. TFS has some cool options for integrating with the builds, etc., but it is definitely not ideal for regression.
TestRail is definitely designed more for Regression testing. To start, Testrail, has two, organization structures, i.e., repository types ypu can choose from, single repository and multiple repository. This is where the organization of it comes into play. Ours was set us as multiple repository when I started working here. It consists of test suites. What i found was that when i created test runs, it was less than smooth to add specific test cases and/or entire groups of test cases to the test runs.
So I changed to the default recommended single repository which has sections and sub sections instead of testsuites. It works so much better.
Basically, what I do is I confirm with upper management which areas and functions of the software are priority using priority 1- priority 4.
We have monthly releases. So each month, I create a Milestone in Testrail for that months release. Then I create a Test Run within that Milestone containing the priority tests divided by Priority. Then as we are testing during the 2 week regression cycle testrail will give you a visual chart showing progress of each run, including ran, not ran, passed, failed, ect. What I like best about testrail is that you can drag and drop the test cases into other folders, you can bulk edit specific data on the test cases at once like priority, etc.
Also, when I first started using it, I created 3 different projects because I started over a couple of times and wanted to see how it would flow with different setups. One other thing is if you edit a test case that is actively in a test run, it will update the test case that is in the run as well, which is nice.
Theres also a plethora of reporting that can be done and can be emailed to you or to a group of people and can be scheduled to regularly be sent.
Hope this helps some!