In addition to what the others have suggested, I'll offer this.
Much of what happens in a fine dining kitchen happens in every kitchen. What doesn't happen as often are inattention, laziness and sloppy work.
So most of all, you should learn to care a great deal about very small things.
In general, train yourself to work clean and neat at all times; yourself, the immediate area you work in as well as the entire kitchen. Be just as quick to grab the broom or the cleaning tools as you are the knife. Leave no evidence behind of the work you have done.
Be concerned that all items are labeled and dated, properly wrapped and stored carefully. Remember the floor is not a trash can. Clean as you go from the moment you begin your day.
Whatever station you work, make sure your mise en place is tight. As you learn your station in any kitchen, learn how much or how little of each thing to prepare. Make sure the quality of every single ingredient is up to par. Keep the insides of your refrigerated units and cupboards as clean as the outside.
Where you are now is as good a place as any to learn to work with grace under fire. When the tickets come rolling in, learn to keep it together under pressure, produce and present food to be proud of, don't slack off or send out an inferior product because you are busy or in the weeds.
Learn how to work professionally with the rest of the staff when the pressure is fierce and you feel like you are losing it.
Learn to handle the sore back, sore feet, cuts, bruises, burns and scrapes while feeling exhausted, sweaty and grimy. Learn to do this with a smile.
Show great respect for your coworkers, all of them ( Every single member of the entire restaurant staff). Don't be that guy. Be on top of your own work and quick in helping wherever needed. This includes every area of the entire restaurant.
Show great respect for the tools you work with. This includes not just your knives but every spoon, spatula, strainer and the stoves and ovens.
Someone paid good money for those tools for your benefit. Show that you understand and appreciate that.
Keep an open mind. Be constantly open to learning new things. As many techniques as you may study, you will be shown what the chef wants you to do before you are allowed to do it. Keep your mouth shut and do as you are told. Each chef may have a different way of doing things. Be sure you understand what that is. When the chef is explaining something, he or she is not concerned that you may already know it. He or she is concerned with how much you are listening to every word he or she says. When you get corrected, nod and say, "Yes, Chef".
A fine dining kitchen may have a fancier stove than the one you use now. The food products will be higher quality and more expensive and perhaps more exotic. There will be more advanced food preparation and plating techniques. There will be more kitchen employees and more going on than you may be used to.
The important part of adjusting to high end dining is to show how much you care. You show this by your behavior. That does not have to happen in a four star kitchen. It can start right now.