Even painting all the trim white is a tremendous amount of work. If you want to stick with stain and want to change the color and want to see the grain through the stain clearly, you really have no choice but to remove the existing finish, either chemically or mechanically (by sanding).
One slightly-easier option (as noted) is to put on a finish that itself contains stain (like Polyshades). Polyshades is polyurethane with stain in it. The stain does not penetrate the wood but reside on the surface. That is good for woods that don't take stain well (particularly ones that get blotchy like pine) as you get a more even coat. The trade off is that you do not see the wood grain as much (because you are covering up with some pigment). Along similar lines, they make semi-transparent glazes that reside on the surface of the wood that can alter the color while still looking like stain. I've used General Finishes glazes before with pretty good results. Either of these options will let you change the color of the existing wood while still having the finish look like stain (but it will be dark and while you will be able to see the wood grain, it will not pop out at you).
Under any scenario, it is a huge project. I am presently replacing all of the trim in our house and it's going on over a year.