Regular drill bits will work just fine. Spritz with a lubricant like wd40 to make it easier.
Copper is fairly easy to drill. A press will make it safer and easier to keep in place. Dremels will drill copper pipe perfectly, and you can also use the cutting wheel to cut slices out of it. I admit, I do think they are worth the money.
I have never done crafts with them, but I do cut and drill copper fairly regularly at work. Love the dremel, but prefer the cordless models. One less thing to worry about. Otherwise I use a DeWalt either 18v or the newer lighter weight brushless 20v one.
Honestly, I'm sure your drill would work just fine.