The best order for interior painting (walls, trim, windows, etc.)

If you must paint this stained wood, a light sanding with 4-O steel wool or 320-grit sandpaper should be fine. All you're doing is roughing up the surface coat, usually poly, which will let the paint "key" to the surface better.

Be aware of the finish originally used on the wood; only poly will take a latex topcoat well, while on most other finishes like varnish or oil-based finishes, latex will not bond to the surface properly; it may peel easily, or even bead up when applied. If this is the case, roughing up the topcoat is not enough; you will have to sand off the topcoat to get to raw wood, and on a profiled wood piece like mouldings, this can be difficult or impossible.

Also be aware of the type of wood; certain woods are "open-grain", and will absorb paint deeply into the structure of the wood, requiring many coats to "fill" the wood and get a solid color on top. Otherwise, the grain and any knots will show through the final coat until you put 3 or 4 thick coats on. If you have to go down to bare wood on an open-grained wood, you have to seal the wood with a product that is compatible with your paint base.

Good question Nick. I manage several rental units and have dealt with the same situations. Originally, flat paints were used in most of the rentals and almost every couple of years they needed repainting especially if kids were in the unit. Flat paint is hard to wash and very susceptible to marring and finger prints etc. The newer flat enamels are better, but not great. I changed over to a harder eggshell finish. Depending on the time a tenant is there, the eggshell really outlasted flat, usually good for 3 to 5 years unless damaged. Satin enamel works great for ceilings, hides marginal imperfections and very easy to clean between tenants. I have also found that since looks is more important than durability, I have used a lot of PVA grade wall paints from Ben Mor. At contractor price, it can be almost half as expensive than premium grade paints, covers well, lasts for several years. I have given up on expecting paints to be show ready after a few years in a rental. Cleaning and touching up walls takes almost as long as painting so a new quick same color wall paint job between tenants freshens up the place and smells new when showing the apartment. I tend to use just a couple of colors for most everything, so I buy 5 gal cans for economy, batch small leftovers and treat older paints with Flotrol. Be sure to keep the color formula in a notebook (ask for an extra sticker) so you can buy a good match later. Changing colors always requires two coats, but one coat of same color is normally fine. You can use your leftovers with new paint if you strain it through a mesh colander and batch it with your new paint, thus slight color changes disappear.