So I have smoke coming out of my exhaust.  I can't really tell if it's blue or black, but since the car is losing oil and there's none on the floor, I'm guessing it's burning oil.  When I first turn on the car there is a cloud, and there is a nice puff everytime I down shift or excelerate.

There is a bit of work involved ut not rocket science. You most likley will have to do some honing to the cylinders. But there may be alot of while I'm in there...........

You can re-ring with the engine in the car, but....why? Ken(ksyter) swapped the crank out of his car with the engine still in it...on the ground...in his yard!
My opinion is, I would pull the engine and do it right. There's no telling what else might be ready to take a doo-doo on it. The cylinders need to be measured for out-of-round & taper, as well as bore size, and then compared to the size of the pistons. It may be that the cylinders are too large(bore size) for the pistons. Cheap rings will cause it too, but normally won't live 5 years before having a problem.
I'd say pull it and start over...just my .02.....

Well, five years doesn't really explain it.  It was in storage for a few years, then it took over a year to paint, etc.  So I figure that I have less than 200 miles on the car since the rebuild. 

From what I'm reading, it appears that this is more than I can do. 

 

Well now...that changes things a bit.
It could be that the rings have never even seated from the rebuild. Depending on what type they were, it may take even more steady driving to get them seated properly, if they even will at this point. Fresh engine re-builds will usually burn/use a little oil until the rings get seated...
Some quality moly/chrome moly rings take a long time(relatively) to seat properly, and even that depends on the quality of the surface in the bores, as in how they were finish-honed.
I might suggest at this point having a leak-down test done, and compare the leakage across the board on all cylinders. If one or two leak considerably more than the others, then it would be best to pull and start over. If they all leak about the same, then it may just need some driving miles to seat the rings better.
Does the engine run ok, other than the smoke?

Thanks for the info Joel!

Slim possibility that it has some damaged, or even missing valve seals...

I would have to say it's the valves leaking down, if it was the rings it would smoke all the time. The valve guides are probably bad or harder than woodpecker lips, this allows oil to drain down through the valve into the cylinder this gives you the blue smoke on start up, when you let up off the gas to shift the engine will suck through the guides and this causes the puff of smoke when shifting. You can change the seals in the car. My two cent's.