If your automatic transmission isn't behaving and needs fixed, welcome to the club! It seems that we all go through this ritual sooner or later, no matter how much you baby the car or how carefully you maintain the transmission. At this point you have two choices: rebuild, or replace.
Rebuilding is usually the way to go with a Ford transmission. This is because many of the parts used in rebuild kits are better and stronger than the original Ford parts, which means increased longevity and better performance longer. This isn't a do-it-yourself situation unless you've had extensive experience with rebuilding transmissions. So you'll need to find a reputable transmission shop or two and get price quotes. Expect the cost to be…well, not cheap. If you believe your transmission is worth saving then that might just be the price you have to pay.
Often, though, you will be offered a pre-rebuilt transmission. That's okay as long as it comes with a good warranty. But you may need to compare that warranty vs. the one for having your transmission rebuilt, and see which one is the better long-term deal. Generally we don't have a problem with pre-rebuilt transmissions but because they're already finished, you don't really know what kind of job was done internally. Whereas a shop can rebuild yours with their own preferred kits, and sometimes they'll even let you see the old and new parts for comparison.
Or you could take a chance and find a used transmission, whether from a private seller, eBay, Craigslist, or a salvage yard. Please bear in mind that you will be taking a great risk on this, as you truly will not know how (or if) the transmission will work until it's installed and the seller has his or her money already spent. You may end up with a gem, but it could also be junk.
Your choice all comes down to money, and the best deal you can get for it. You'll have to do your homework here.
Visually you'll be fine, as a Ford casing and pan will be supplied either way (via reuse or a core).
The temptation to have a shift kit and/or a higher stall torque converter may be presented to you. A very mild shift kit actually helps improve shifts a bit, with less slippage and therefore less heat, so that's actually a good idea if possible. A higher stall converter might be okay in conjunction with the shift kit, but we'd recommend no more than 100-200 rpms over stock (which is usually in the 1500-1800 rpm range). Don't forget, the converter has to work along with the cam, so going with too high of a stall speed may bog your car down. Be very careful here.
For the Ford AOD transmission, a new performance valve body could also help improve shifting and performance. While they can work in conjunction with a shift kit, they should be installed alone for easier service. Again, this is an added expense but may help solve any issues you've been having, plus you will likely enjoy the snappier shifts. There may also be C5 kits available for all cars equipped with such transmissions, as the C5 and popular C4 were very similar.
Fluid should be at least Dexron/Mercon (originally at version II but at this writing it was up to IV, which can be safely used retroactively). Aftermarket performance fluid should be alright so long as it doesn't void or otherwise interfere with the warranty on the rebuild/replacement.
An aftermarket transmission cooler is a good idea, because heat is the number one killer of transmissions. You do have a factory one, sort of: the side tank on the radiator has hookups for your factory transmission lines. It's not the most efficient way of cooling the transmission fluid but it's better than nothing. Your stock lines can be adapted to go into an aftermarket cooler, which can then be attached to your stock radiator or a/c condenser. It's good insurance but not a necessary thing if you're just going to occasionally drive the car, or if you're not going to be running at high speeds.
Please be very adamant that your stock kickdown cable/lever and all brackets are reused! These parts are very uncommon these days and are nearly impossible to find new. Lokar does make a generic kickdown cable that can be modified to use in our cars…it's about the best and only solution out there now, as the original Ford cables are unavailable now.
One last item: especially with the AOD, there were two versions: standard tailshaft and long tailshaft. The longer tailshafts were used in Lincolns and some Ford trucks. Be sure to specify that you need a short tailshaft version, as that's the only kind our cars had. Otherwise you'd be looking at a new driveshaft getting made, which adds to the overall expense and will delay the return of your vehicle.