The PST can't sync while Outlook has it open but the service will keep trying, often creating a lot of small files which waste your storage space. If you use an online service, such as Dropbox, OneDrive etc, to transfer the PST file, you should copy the PST to a folder outside of the sync service. You can do it the old-fashioned way, using a "sneaker net" and transfer the pst on a USB drive, or access the other computer over your local network and copy the PST. To copy calendar and contacts or archived mail to a second computer, copy the PST data file between your computers. You can send appointments and contacts to yourself as needed, or use one of the methods below to copy non-mail items to your other computers.
If you use a POP3 account, leave mail on the server until the other computer has downloaded the messages and BCC yourself on sent messages.
For simple "syncing" of email, choose an IMAP email account if your email provider supports it.