If I have created a project but 2 different teams will be working on separate boards, how do I separate out the tickets so only the relevant ones show on each board instead of all of the tickets in the project?
Boards are not containers for issues, they are a selection of issues (defined by a filter)
So, to get issues on different boards, make the boards select only the ones you want on each board.
If your first board selects for a project, it'll have a filter like "project = xyz". So change that to something like "project = xyz and issuetype in (bug, feature)" and then create a second board with a filter like "project = xyz and issuetype not in (bug, feature)"
To add to what Nic is saying here, how you separate issues between your teams will ultimately define your boards. For example you might, as Nic has shown use issuetype. However you could just as well use components or labels or a custom field.
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