I think I’d be looking at an external audio interface. Something like a Presonus C24 or a Focusrite 2i2 or similar or better.
I would go with Tony's recommendations. I love and use the Focusrite stuff and Presonus always has good reviews. Use the Google to look up the manufacturer sites.
Do I additionally need an internal sound card for other things that are related to the computer?
No, the external card will handle all of the computer's sounds. The sound card will usually connect to your computer via USB port.