We are thinking to use lightning action, as it's awesome in lightning experience. But in our case we have to still support classic users.
Can someone confirm this that a lightning Action will work in classic or not?
You can wrap your component in a visualforce page, that you can then use within your classic interface, with close to the same code.
Look at Using Lightning Components in Visualforce pages
The only subtlety is that you have to handle either app event firing or just the e.force events in either experiences: Classic or Lightning in your code to make it generic and the same code base.
I assume that you are talking about actions. Actions are not specific to lightning.
They were called Chatter actions before. In Salesforce Classic they will appear in the chatter feed input field.
Lightning actions
While Salesforce Classic displays actions in the Chatter publisher, and buttons on a record’s details page, Lightning Experience blurs the distinction between these elements. The new interface intermingles actions and buttons, locating them in different areas, based on function.
https://help.salesforce.com/apex/HTViewHelpDoc?id=lex_find_actions_buttons.htm&language=en_US