You will need to edge band the exposed edges---filling/etc is a terrible p0rocedure.

Luckily, there is iron-on edge banding---and HD probably sells it. If not, maple edge banding is available at woodworking stores and some lumber yards.

Using plywood is actually better than pine---the plywood will sag less. And, if you were to add a 2" wide skirt on the front of the shelf, the normal sag would be minimized further.

Glue and screw a 1x2 across the front of the shelves.

It finishes the edge quickly AND stiffens the center front of the board so it's less likely to sag.

Thanks for the suggestions. I like the idea of stiffening the shelf. Will finish nails suffice? Screws seem like it be a pain to fill over

Finish nail will suffice, but you should also glue the front edge piece to the plywood. I will be using a nailer and probably 1.5" 18-gauge brads when I get around to doing mine. The nails are mostly just to hold it until the glue dries.

I've had very, very, very good results with iron-on edgebanding. I applied them with my household iron, and then burnished them w/ a block of scrap pine while still warm. I even purchased a squeeze-together trimmer to clear away any excess. Then painted them white.

It was fast. It was easy. I covered a ton of shelving--let's see, five shelves on each side of the living room, each shelf about 5 feet long. And it still looks good. And I still say--it was fast.
And just not very fiddly.

For the closet, I'd go that route.

If you want to apply board to the front as a stiffening I'd feel like I'd need to clamp in addition to nailing. Maybe you don't--but then I'd be worried I didn't get a really good bond.

I have books on my shelves, and the plywood is strong enough--it didn't need stiffening. I have supporting brackets every 2.5 feet or so.

If you're just storing clothes, etc., even in boxes, you probably don't really need to stiffen the front anyway, depending on how well-supported the shelves are.

trimmer for edge banding--simple!
http://www.veneersupplies.com/products/Edge-Banding-Trimmer.html

The squeeze-together one I have (though I wish I had the one above!)
http://www.mcfeelys.com/product/156044/FastCap-Fastedge-Edge-Banding-Trimmer

Here is a link that might be useful: The Sagulator