Hi, Saddlebum. Before making a recommendation, there are a couple of important points to sort out.
Firstly, for dove and quail, a longer barrel is usually used (26" sometimes, more often 28" or even 30"). For turkey, a shorter barrel (often 24") is more frequently encountered. You'll have to work out which type of hunting will be your dominant activity, and choose from there. Personally, I'd suggest a 26" barrel as a good all-round choice.
Secondly, the loads you shoot will differ very greatly. For dove and quail, you're talking small shot sizes, and probably 2¾" shells. For turkey, you're talking about 3" or 3½" Magnum shells with much larger shot. If you plan on some clay shooting, you'll be down to low-base, low-charge target loads. Thus, you'll need a gun that is able to accommodate the entire range.
Given these factors, if you want to do them all with one gun, I'd suggest the Winchester Super X2, which is available at around your price range. The Browning equivalent (they're basically identical shotguns) would also be good. If you can find one at a good price, I've also heard very good things about the Fabarm semi-auto (Fabarm is the shotgun division of H&K, which already speaks volumes for their engineering and reliability) - I haven't used one of these myself, but I've handled one, and am very impressed by it. I'd recommend against the Remington 11-87, as its quality control and reliability have been very spotty in recent years.
However, you might be better served to buy two shotguns. Make one a dedicated turkey gun, with shorter barrel, camouflage finish, and a 3" or 3½" chamber (personally, I find the 3½" shells unnecessary and too much to handle recoil-wise - I've never used or needed more than a 3" shell). For dove and quail (and occasional clays use), consider a lighter gun with a longer barrel. Here, the Remington 1100 might be a very good option - it hasn't had the problems with quality control and reliability that the 11-87 has had. There are also good options from Benelli, Beretta, etc.
That's my take on the subject. YMMV...