Setting up future meetings with participants using Teams calendar or Outlook integration
The reason you are not seeing the breakout rooms option is that Microsoft 365 Personal does not include the ability to create or manage breakout rooms in Microsoft Teams. This feature is available with eligible business and education subscriptions.
You do not need to purchase Microsoft 365 Business Premium just for breakout rooms.
Microsoft 365 Business Basic (with Teams) is a suitable and more affordable option that includes this feature. Microsoft 365 Business Standard and Business Premium (with Teams) also support it. Basic doesn't include desktop apps for Word Excel Powerpoint, etc though.
You can compare the plans here:
https://www.microsoft.com/en-us/microsoft-teams/compare-microsoft-teams-options
Please note that Microsoft 365 Personal and Microsoft 365 Business are separate subscriptions. Purchasing a business plan will require a work/business account rather than simply enabling additional features on your existing personal account. If you don't have one, it will allow you to create one during the sign up process and the domain ends with onmicrosoft.com. Moving forward that's the account you use, not your personal account email.
Once you have the appropriate subscription, sign in to Teams with your business account using the desktop app on Windows or Mac. You should then be able to create breakout rooms as the meeting organizer.