Assuming you only care to hear from a handful of people, open your phone’s Settings, and tap on Notifications. From here, tap on Do Not Disturb and toggle it on. Now, you can tap Allow Exceptions and add the contacts you’d like to get calls from.
In the Phone app , Visual Voicemail (available from select carriers) shows a list of your messages. You can choose which ones to play and delete without listening to all of them. A badge on the Voicemail icon indicates the number of unheard messages.

Generally, people that call and leave messages are more likely to be ideal customers of your business. Your voicemail message is often not much more than a formality to the person calling.
Of the many things your phone can do, automatically recording a phone call is not one of them. Perhaps not including that feature is a security precaution for phone makers; it could be a potential liability issue. Regardless, from time to time, documenting a call is a useful thing to do, whether you’re conducting a phone interview or cataloging a customer service call.