First off I want to take my hat off in admiration of the job you have done. Considering all the work it took getting to where you are I would incourage you to make your own decisions on staining and top coating. It's obvious your are not lazy and looking for quick fixing the project. As a matter of opinion and to give you something to consider. The room would look good with the piece finished "primitive",and the piece looks like it would agree. Going that route benifits if some old finish remains. I would sand it smooth, either clear coat as is or use a light wash coat of light color translucent stain then clear coat. getting the best look requires standing back and allowing your eyes and taste guide you in wherther to sand,stain or scrape here or there. One simple but potentially impressive trick is sanding slight recesses in places where years of use would have resulted in wear such as next to handles and latches. Do the deep sanding AFTER WASH COAT for best effect. The really adventurous will make a few gouges and dents then fill with finish same color as origional if non are naturally present. The old pros will run down DIY product lines and reccomend professionel grade products. If one is inexperienced it's best sticking with DIY products until they pratice with pro grade a few times on scrap boards. Whatever you choose,try to stay with a single system from start to finish,following directions on containers. Min-Wax and Deft are easy to find products. Good luck,have fun, trust your heart,judgment and eyes.

Your almost halfway done, if you plan on a clear/stained finish. If you intend to paint, you're about 95% ready.
In any case, mind the lead dust, take every precaution.
Casey

Thanks everyone! I'm leaning toward the stained finish. What kind of sanding techniques do you recommend (type of sander, grit, etc.)? This is my first piece and I'm a little nervous about sanding it in such a way that will make the sanding marks obvious after I apply the stain. Thanks!

When I strip paint, and it's a little further along than the face of yours, I put another coat of stripper on it, let it sit, and then scrub it with fine steel wool, at least 00. Wipe it off and then wipe it again with a mineral spirits. If there is any paint left on at that point, I apply a coat of mineral spirits and go over again with steel wool. If anything left after that, those spots probably need more stripper.

What kind of stripper did you use? I've had good luck getting down to bare wood with Citristrip. Put it on more like frosting than paint, cover it with plastic to keep it wet, and let it sit for 24 hours - no peeking! The remaining paint should scrape right off with a plastic scraper. Rinse with water and a plastic scrubbie.

Here is the bottom of a table leaf after I stripped the top, but before I did the bottom - some stripper went over the edges. You can see at least two layers of paint.

The bottom of the same leaf - after. The oak grain is so deep, I couldn't get all the paint out of the pores (Lord knows I tried!)

Yes. I like to do the same as above, another coat of really good caustic stripper, scrub it off the steel wool and clean with minerals spirits.

By then you should be ready for sanding. I usually use an orbital sander somewhere around 120 grit and then hand sand at 220 (just a light sanding, you don't want it too smooth for staining).

Unless you want to stain it very dark, avoid using caustic stripper (high pH like lye and Peel-away 1) on pine or oak because they make a permanent color change; oak turns dark brown, pine will turn green. Since it has to be water-rinsed/neutralized, it also raises the grain severely. i would use a methylene chloride stripper like zip strip or Kutzit.
Casey