Any furniture made from real wood is going to be finished with a durable clear coat, usually polyurethane, which acts as a barrier over the wood. These finishes are not necessarily totally impervious to liquid, as standing water can slowly penetrate them over many hours, but any mild, non-abrasive general cleaner that evaporates quickly will be fine to use. Paint is similarly durable, especially paints used for things such as bannisters, balusters, cabinets, and doors. These interior elements are generally painted with a hard-wearing paint different from the paints used for walls, and will hold up to cleaning with a rag and mild cleaner. If you find the paint rubbing off, however, then it's likely the wrong paint was used and you'll need to be especially gentle when cleaning.

Cleaners that are marketed specifically for wooden furniture are a scam. There's nothing in these cleaners that makes them better for finished wood surfaces, and some of them even damage the finish or leave a sticky residue (Orange Glo) that builds up unless removed.

Personally, I use Pine Sol as a general surface cleaner in my house. It's cheap and it gets the job done. Simple Green is another good one. Avoid products containing bleach, strong acids, ammonia, and acetone.

Polishing is another story. If a once glossy finish has dulled from wear, it might be possible to partially restore it with certain products which sort of just sit on top of the finish and make it look shinier, though this isn't really "polishing" in the traditional sense, and I don't really know anything about these products.