How can I make this into a wine/beverage cart?

You could paint it a color complementary to your home, then add castor wheels to the bottom and a mirror to the top. Good luck!

Casters is a good idea but you might want to put it on a base first as those legs might not be able to take the kind of strain. I think the center shelf would make a great spot for a big ice bucket! The bottom might be an opportunity to see if you could build in a wine bottle holder.

The whole piece looks like it might make a nice opportunity for a paint and stain combo. You can paint out the whole thing but sand down the top and stain it in a rich color.

You could get some ideas from these.

https://www.hometalk.com/search/posts?filter=beverage%20cart