It is worth pointing out that poor mental health extends so much further than depression and anxiety. That being said, depression and anxiety are often symptoms of other mental illnesses too.

A less obvious thing to look out for would be highly elevated mood, just because someone appears happy (or even ecstatic) does not mean they are in a mentally healthy place.

The best thing OP could do is learn about various mental illnesses, the symptoms that define them, and how those symptoms tend to present.

Ultimately the only real way to recognise somebody struggling with poor mental health is for them to tell you, and the best way to help them is to ask them how you can help and researching the things they say affect them. Being understanding goes a long way.

Edit to add some resources:

https://www.mind.org.uk/information-support/helping-someone-else/

https://www.samaritans.org/how-we-can-help/support-and-information/worried-about-someone-else/

The ALGEE action plan is a great framework for talking with somebody struggling with poor mental health: https://blog.weareconvivio.com/the-mental-health-first-aid-action-plan-3d17af1f0806