Bard doesn't really have a "rotation" as it's more of a priority system.

Make sure to use Wanderer's Minuet. Always cast Straight Shot first, followed by Windbite and Venomous Bite. Use Empyreal Arrow and Blood Letter when available. Iron Jaws should be used when Windbite and Venomous Bite have around 3 seconds left. You shouldn't be losing these DoT's. Sidewinder is your main damage dealer, use this after applying your DoT's.

Obviously weave in cool downs (Blood for Blood, Raging Strikes, Hawk's Eye, Internal Release) and use songs as necessary. You can also use Repelling Shot and Blunt Arrow if the situation allows you.

I've probably forgot a few skills as I don't have access to FFXIV right now but those are the basics and should allow you to do significant damage as a Bard.