I wonder if the free upgrade my premium account I can use for my work gmail account and my personal gmail account to have everything synchronized on the same task manager.
Unfortunately right now Todoist can only be used with one email address, that is only one email attached to the account.
You can add separate email accounts to the list of accepted addresses which you can email to create tasks. You can read more about it here: https://todoist.com/blog/2013/01/zero-inbox-policy-with-todoist/
There is no separation between work and personal, unless you want to use Labels to distinguish between them. It is just one account at the moment, no multi-sign in as of now.