Cleaning Wooden Floors...??

I used to work for a flooring contractor and can tell you that Bona flooring cleaner is recommended by most flooring manufacturers. It is made specifically for wood floors. You can purchase it at OSH and most hardware stores and from flooring contractors who have retail space.
NEVER use Murphy's Oil Soap or steam to clean a hardwood floor. No reliable flooring contractor would advise it. Steam can cause the wood to swell and warp. MOS leaves a residue behind that ruins the finish in such a way that when your floor needs "refreshing", you'll be paying for a complete refinish job, not just a "screen & coat". A complete refinish of a floor requires sanding down to bare wood and applying multiple coats of finish. A screen & coat is a light sand and application of a single coat to refresh the finish, which costs about half as much (or less) as a complete refinish. If you're really in doubt and want to make certain you do the right thing for your floors, contact a local flooring contractor and get a recommendation. Bona does a wonderful job of cleaning and enhancing the finish on your floors. It's really the best way to go.

I use Murphy's Oil Soap. It is recommended for wood. You can get it at most grocery, hardware, or drug stores.

I use a steam mop on my hardwood floors.

For mopping, Method has a hardwood floor cleaner (check Target) that smells really nice and does a good job. Just make sure to get up as much of the water as you can so the floors don't warp.

First are they hardwood or laminate.

Hardwood you can use a steam mop on, most manufacturers of laminate do not recommend a steam mop, it can cause warping.

Second: If they are hardwood, are they sealed with a top coat? If so you are not cleaning the wood, you are cleaning the finish. Think of your finish as a mirror or window. Taking Murphys Oil Soap to a mirror will leave a cloudy mess

Use a natural broom to remove debris, and mop with vinegar/water solution. You can add essential oils if you'd line. The mop should be damp. Polish with a dry microfiber flat mop head, you can spray the microfiber with windex.

Laminate can be done the same way, use a damp mop, you should not be able to squeeze any access water out of the mop. Then polish with a dry microfiber flat mop.

Vinegar and water work well for me. I also use the Bona spray for my wood floors.

I have read that over time, the vinegar may dull the finish on a wood floor. So far ours is fine.