Which test cases/tests should I run?
No, running Drupal core test suite is wrong. You can assume that your code won't messed up drupal core and if you so - you'll get exception for that and then the fun will start :).

If you came from the ruby world then you probably familiar with the Behat solution and if not you should check this - very easy to use QA platform and a lot of posts were written in order to test your site with behat. But there is also selenium that can help you.

And i'll give a you a tip on how to start the testing suite for your site: For now you probably have a running site. You need to take a pen and paper or a google doc and start what features you create for your site: what happens when the use login, what happens when he create node. After you wrote your tests you can start to write them in test - behat, selenium simple test... what ever! As long as those tests testing your features.

Another tip is: when ever you have a bug and you fixed this, you write a test that verity that the bug can't be reproduced. This the way all of the big projects i know works: OG, Message stack, Open scholar,