Assuming you still love it, you could:

-set it on a smaller table (one which it covers the entire top of) and use it as a bedside table.

-store it (hung on the wall even, like you suggested) and take it out as needed to hold glasses for large parties, carry drinks to the backyard, clear up after a party, etc.

-set it up vertically behind a display in a bookshelf.

-use it to create a table to sit next to a rocker on a covered porch.

-place it on your dresser top to corral all your pretties. The oversized look is "in"!

I find that if there is a piece I love, it will cycle in and out of use and I'll find new ways to use it every few years or so. Yours isn't a huge piece, so it could even slide under your bed or dresser or tuck in the back of your closet until the next time it cycles into service.

If you don't love it, maybe this is a good time to donate so someone who will love it, can!

Yes, I do in fact love this piece and want to continue to be able to display it. If I hung it on the wall... how would you suggest I go about doing that? Any tips so it's not just a tray on a wall? Should I put something on either side of it? Or place something inside of it?

Try it on other tables in your home or on a buffet. If you have a fireplace, try the hearth. Use it on a table (with large pillar candles) outside in the summer or use functionally to bring food/dishes etc. outside in the summer.

Check to see if it contains magnetic metal in addition to copper. If so, after you hang it in your kitchen you could use it with magnets to hold up lists or coupons, or in a different room to hang unframed art.

You should be able to find a plate hanger that will accommodate it for hanging, and if the hanger doesn't come in a copper, brass, or bronze finish spray paint it!