What is the best way to start conversations about mental health with my teammates in a lighthearted yet caring way?
I think that the best way to talk about your mental health with your teammates is to be really transparent. In my experience, the clearer you are about what you feel and what you need for support the better, 100% of the time. If your team has captains or someone in a leadership role that you trust (this could also be a coach!), ask them if you could set aside a time during either team bonding, practice, or whatever works best to have a super open conversation regarding mental health - but be yourself! You don't have to make having a mental illness or struggling with your mental health seem super grave and intimidating. Keep your personality incorporated into the discussion and just be super honest. The more your teammates know about how they can best support you, the better the experience will be for everyone. You can talk about your triggers, when you feel the most vulnerable, anxious, depressed, etc (and when you feel the least!! This is suuuper important!), and what you are looking for in support from your team.
Having this be a team-wide discussion also opens up the floor to your other teammates who may also be wondering how to discuss their needs for keeping their mental health in a good place! When I was a junior in high school, I remember my coach sat us down in a big circle and just said to us, "So what do we know about anxiety?" and it opened up a HUGE discussion about the importance of mental health. By the end, almost everyone had shared. All it takes is for someone to break the ice - even if that means starting the conversation yourself if your team doesn't have captains or you don't have a coach you trust.