I am new to microservices architecture. The design principle says a microservice has its own database. Should I be worried about the security of connecting to the database or the security of the communication between microservice and database?
Yes, that traffic should be protected with TLS or some other method.
Microsoft lays out some thoughts an options here (even though they steer you towards MSFT products, the concepts are relevant).
Best practice: Secure access from multiple workstations located on-premises to an Azure virtual network.
Detail: Use site-to-site VPN.
Best practice: Secure access from an individual workstation located on-premises to an Azure virtual network.
Detail: Use point-to-site VPN.
Best practice: Move larger data sets over a dedicated high-speed WAN link.
Detail: Use ExpressRoute. If you choose to use ExpressRoute, you can also encrypt the data at the application level by using SSL/TLS or other protocols for added protection.
Best practice: Interact with Azure Storage through the Azure portal.
Detail: All transactions occur via HTTPS. You can also use Storage REST API over HTTPS to interact with Azure Storage and Azure SQL Database.