(1) Background: In Australia, young people are one of the most vulnerable populations to the mental health impacts of climate change. The aim of this article was to explore mental health promotion issues related to climate change for young people in Australia. (2) Methods: An exploratory mixed-method approach, co-led by young people, was used to engage young people living in Australia aged 18–24 years in semi-structured interviews (n = 14) and an online survey (n = 46). Data were analysed thematically and with descriptive statistics. (3) Results: Findings indicated that negative impacts included worry, eco-anxiety, stress, hopelessness/powerlessness and feelings of not having a voice. Several mediating factors, in particular social media engagement, highlighted the duality of mental health impacts for young people’s mental health. Positive impacts of climate action included feeling optimistic and in control. (4) Conclusions: This exploratory study contributes to an emerging field of public health research on young people’s mental health in a climate-impacted Australia. Climate change is a significant concern for young people, and it can negatively affect their mental health. The findings can inform the design of public health interventions that raise awareness of climate change-related mental health issues among young people and promote their participation in nature-based interventions, climate action and empowering social media engagement.