I want to sync Outlook and MS Project on my laptop. I have Outlook 2010 and would buy MS Project 2010 if they can sync tasks. Do they require Sharepoint to sync if they are on my computer?

You need to buy project server and professional versions.
The latest version is 2016 as 2010 is no longer officially available.
Further details here: https://social.technet.microsoft.com/Forums/projectserver/en-US/d515887a-c9d0-4524-8c9e-80d41f20d808/what-do-i-need-to-sync-tasks-and-progresses-between-ms-project-pro-2010-and-outlook-2010?forum=projectprofessional2010general