Depending on the phone you have can vary whether or not you have to. Apple phones should be good whether backed up through iCloud or not, certain Samsung phones the backup is connected to the SIM card.
If you are using an iPhone, your contacts can be stored in the iCloud so that they are synced across all your devices in the ecosystem. If this is the case, you should be fine switching out your SIM.
Is there anything I need to do before switching out my SIM card so I don’t lose any information from my phone? Sounds like I should just import my contacts to my phone and should be good but wanted to make sure I’m not missing something.
It wouldn't be bad idea to backup your contacts, and user content (such a pictures) before inserting your Public Mobile sim card. Although this is not supposed to happen, Samsung phones are noctorious for peforming a factory reset without warning after switching sim cards.