Hi stojef2014,
I see from your post that after updating to iOS 13 on your iPhone Mail will open to the last opened email and Contacts opens to the last viewed contact. I can see why you'd reach out about this behavior; I'd like to help. First, I'd like to clarify the behavior you're seeing; for example, in Mail if you view a message, return to your Inbox or another folder, then switch to a new app or the Home screen, the next time you open mail you see it return to the message you read, but had exited from?
For Contacts, are you returning to the alphabetical list after viewing the contact you need and having the same behavior happen as described for Mail?
If so, I'd recommend trying these steps first:
After restarting take some time to test for the behavior. If you see the issue continue, I'd recommend creating a backup: How to back up your iPhone, iPad, and iPod touch
Then use these steps to remove and redownload the Contacts and Mail apps one at a time to see if the issue persists: Delete built-in Apple apps on your iOS 12 or later device or Apple Watch
Lastly, if the steps above don't help with the behavior, and you haven't already updated to the most recent release of iOS 13, please do so: Update your iPhone, iPad, or iPod touch
Best Regards.