If you are using a date column for birthdays you can create a calendar view directly on that list.
If you want to create multiple calendars you can stack them together (overlay them like in Outlook). This will only work for Calendar Apps though - not list calendar views.
- To do that open your calendar.
- In the ribbon select the "Calendar" tab and find the Calendar Overlay button.
- Add in the URL of your other calendar and select a color and there you go.
The simplest method is to create a view in your list showing just the name and birthday, then copy that data directly into your birthday list. You may also want to consider just creating the Calendar App and adding columns for your other data, and just moving all the data there if that is the primary purpose of the list, that remove one layer of duplication.
The next step up is creating a workflow on your list that copies that data. You can build it with SharePoint Designer and they are not too difficult to do, even if you have not used them before. This would require you update each list item or manually start the workflows on each item though to get it to work.
And the most valuable - automating it all, is the toughest if you do it your self because it requires varying levels of development effort, depending on which route you take.
If you are configured to support "apps" I would consider looking in the Office Store to see if there are any there that might also meet your needs. I have not been thru there in a while but I would be shocked if there were not 3 or 4 different birthday / anniversary apps out there.