Yes, DNS servers would be expected to contain multiple records and operate as a standalone service. They are the databases of the domain name system. Plural, because more than one is expected for availability of this critical service.
Server Fault does not do product recommendations in general, like what DNS software to use. However, given your lack of an existing DNS infrastructure, I recommend starting with your domain registrar or hosting provider's managed DNS service. Generally inexpensive and higher availability than what you might put on a single virtual machine.
Later, if and when you attempt to create your own authoritative DNS servers, come back with questions.