If you are using the Notes app all you have to do for iCloud support is to turn it on under Settings > iCloud. It will keep all your devices synced. If you have a Mac running the latest OSX, then will you can sync with Notes on your Mac. You can also get to your notes by going to icloud.com on a computer.
Evernote is another good solution. It is far more powerful than the Notes app, and will sync with with just about any device or computer you can think of. And for those occasions that it won't, you can get to via a web browser. For power note taking I heartily recommend it.
The only text/note app I use that can sync with DropBox is Daedalus. It's an unusual text editing program that is based on the idea of stacks of paper/pages. Unusual, but easy to use once you get the hang of it. More of writing app than a note app, it could still be used effectively for notes. It will sync with DropBox, Box, or any other WebDAV server.
There a lots of text/note apps that will sync with DropBox, but I'm not familiar enough with the others to make recommendations.