It feels rough and gets messed up if you don't start at the ends, because your hands push against the exposed edges. That's why you get finishing tape in the box!
Wrap one side mirror-image of the other. Don't forget to cut a little strip to go behind the lever hoods or you will have an annoying triangle of bar showing. Wrap it as tightly as you can, and always ensure the tape comes over the top of the bar towards you on the tops, or you'll unwind it when climbing.
The tape to avoid is anything too cloth-like that gathers dirt. One puncture and my yellow Boardman bar tape was a mess. I bought some smoother Velox tape to replace it.
Old-fashioned shiny tape looks good on vintage bikes, especially those with metallic paint. No cushioning but you can wear gloves if it bothers you.