The newly routed molding should be stained BEFORE it is nailed in place. There are various methods for staining molding. The brush-on method, in my view, does not achieve a sufficiently deep coloration and can leave streaks on the wood. I prefer the old-rag method. Not only does it ensure a deep and even spread of stain, it also ensures you will get stain all over your hands. An oil-based stain will be with you for weeks. This can be a good conversation starter at work, school or at the hardware store. Conversely, people may think you have a severe case of liver spots.
Varnish is a sealer coat that is used over stain. It keeps the wood from drying out and other stains from marring the wood. Make certain the stain has dried before application of varnish.
The old-rag method does not work well with varnish. When varnish gets on the skin, it tends to be sticky, making it difficult to put down a rag, a brush or any other utensil. Releasing any object can be like trying to get a piece of tape off your fingers. When it dries on the skin, varnish tightens up like a clear scab and flakes off as if it were a symptom of some terrible skin disease (shingles maybe). Varnish, then, should be spread with a brush. The size of the brush is dependent on when you want to finish the job, now or next Tuesday.
Varnish comes in gloss, satin, flat and everything between. Gloss is shiny; satin is deep; and flat has no lumps.