Dropped Tags when burning CD

When I burn a CD using Windows Media Player, none of the tags show up on the CD. What am I doing wrong? The mp3 files are my own recordings so no online data would apply.

Do you create a real audio cd or a data cd with mp3s on it?
Tracks on a real audio cd (are a kind of wav file) and do not contain any tags.

I am making a real audio CD that can be used in any CD player. If I want to burn an audio CD that would be just like one I can buy in the store with track, composer, performer, artwork info, etc.....how do I do that?

"Real" cds in the CD format do not contain any tags. And as you have already discovered that there are no tags any more you have probably done it right.
If you want to create a label with the additional information you probably would have to resort to an export of the data from the tags and then print it in a suitable format.

If, on the other hand, you have cd players in mind that can also play mp3s then do not use the WMP to create a CD but use the Explorer (or a suitable cd burning program) and create a data cd that contains mp3-files.

Audio cds do support tags.
It's called CD-Text

The workflow in question (using WMP as authoring tool) does AFAIK not lead to a CD with CD text.

Do you know of an authoring tool that will lead to a CD with CD text?

Read my CD-Text link.