I think what may be best for this process is to list by sets, then by teams for unclaimed sets. I am not sure on how to work this though. I'll start by listing the sets I have a bunch of cards from and approximately how many.
***claimed***2008-2011 Bowman- These can fill up maybe an 800 count box, I do have some from earlier years too.
***claimed***2007 Fleer Ultra- 250 card set, about 200 here, 201-250 are short prints, I have about 30.
***claimed***2007 Topps- I have an extra full factory set. It is open and in numerical order but is complete.
***claimed***Topps Heritage- All pre-2009 sets, about 200 cards
***claimed***Topps Chrome- Mostly 2008 but some from earlier, about 100 or so. Inserts, refractors, etc.
***claimed***Topps Allen& Ginter and Turkey Red- 2007 to 2009, maybe 40 cards between both brands. I also have plenty of 2011 Ginter I can include too.
***claimed***1990-1992 Upper Deck- Mostly 1990, about 400 cards here.
1994 Upper Deck Collectors Choice- Two retail boxes worth, silver and gold sigs included.
2007 Upper Deck- About 700 or so (I'm just kinda guessing here, its a pretty full 800 ct. box)
2008 Upper Deck- About 150 cards.
***claimed***2008-09 Upper Deck Goudey- 40 or so cards between the two years.
***claimed***2008 Upper Deck Masterpieces- 30 or so cards, two framed parallels.
2009 Upper Deck Baseball Heroes- 30 or so cards.
***claimed***2009 Upper Deck Goodwin Champions 30 or so cards.
***claimed***2009 Upper Deck Icons- 100 card basic set, maybe missing 2 cards. Some rookie short prints and inserts.
***claimed***2009 Upper Deck O-Pee-Chee- About 70 or so, with black bordered parallels
2009 Upper Deck Piece of History- 30 or so cards.
2009 Upper Deck Signature Stars- Maybe 50 cards out of 100 card set.
***claimed***2009 Upper Deck SP Legendary Cuts- Very close to a full 100 card base set, missing maybe 15 or less. Also a few short prints.
2009 Upper Deck Spectrum- 40 or so cards.
2009 Upper Deck Timeline- 30 or so cards.
2010 Upper Deck- 125 or so cards.
I suppose cost will come down to how much it will be to ship. You can claim what ever sets you'd like and once you're done, I'll figure out a price for shipping, supplies, etc. (boxes and such can get expensive), prices will be very fair.