you do have to take the polyurethane coatings off and get below the drips and runs. I would do this with sandpaper, starting with medium grit and do sand only in the direction of the grain. when done sanding with medium and below the drips, then switch to fine or 220 grit paper and smooth out the surface, then go to steel wool. Now, you are at a starting place to begin staining. Now, I love minwax stain, but I hate stain with poly in it. This for some people saves a step but with the poly urethane there, the top coat will stain and seal and you cannot re stain for a darker effect. So, go with a stain only and then when there, do a poly seal job.
You do have quite a bit of work to do on this project.