How sure are you that the top veneers are pine? I suppose it's possible, but I can't recall ever having seen pine veneer work.

It sounds like you just need to replace the edgebanding. If the corners are still sharp (i.e. you haven't rounded them over with the sandpaper) then you can get some iron-on edgebanding and stick it in place with a household clothes iron.

Edgebanding is a bit wider than the likely 3/4" thick top, so you'll have to trim it flush and soften the corners after ironing it on.

jon1270 - pretty sure the top is pine - is there another wood species that has large knots? Unfortunately the corners are pretty rounded over. The veneer that was there was already worn off around the edges - they were that way when I bought it. Not sure that edgebanding would work. The top is wider than 3/4". Looks like they took two pieces of MDF and glued them together to make the top. It is probably 1 1/2" thick. I think I'm either going to have to try the stain or paint it. Would wood conditioner help the stain?

Every sort of tree has large branches, which means large knots; it's just that those parts of the tree are usually thrown on the scrap pile or used for something other than furniture.

Can you post a picture or two?

MDF is easily painted. Staining is not a really good option. Also, painting the edges of MDF is different from painting the faces.

However, are you sure it is MDF---medium density fiberboard---and not chip board?

The edges of real MDF looks like many layers of paper and the faces are smooth and featureless.

Chip board is made by mixing wood chips with resins and binders and pressing the mix into sheets---that show the individual chips. Some chip board looks more like pressed sawdust than wood chips.

That is not stainable at all.

This place has wide edge veneer 1.5", although pricey. You can get 7/8" wide veneer locally.

Here is a link that might be useful: edgebanding