I'm pretty much done tweaking the addition of gears to my wife's adult trike. The actual mechanics are covered in this post, this just deals with a final parts list and sources.
For the parts you need, donor bikes are by far the best source--An entire used bike can be purchased for less than the cost of one part.
You will need:
A 5 speed freewheel, preferably Shimano or Falcon. If the bike has index shifting it should work, as will most Shimano Uniglide and most Falcon. Most will have 28 teeth on the biggest chainring--That is enough, but 30 or 32 would be better. Most freewheels with 6 or more cogs will be too wide, requiring excessive spreading of the aluminum frame, potentially weakening it. If you can get an old "ultra 6" freewheel, it should fit, and you can use 7 speed shifters with it. (Adapting these instructions to a steel-framed bike spreading is much less of an issue)
5 or 6 speed index shifters--Preferably Shimano, but most are Shimano-compatible. You can also use friction shifters if you prefer, or most lever shifters have a friction mode. Suntour parts were good, and will work in friction mode. Suntour used non-standard index spacing, so aren't likely to work well with other brand parts.
Dérailleur chain--the existing chain on the trike is too thick for dérailleurs.
Dérailleur--Should be index compatible if you want index shifters. Shimano is the best choice, but there are a lot of clones of the basic Shimano Tourney derailleur--almost anything "index" that isn't Suntour will be Shimano-compatible.
Miscellaneous cables--For this application, you'll want a length of cable housing from the shifter to the derailleur, since there are no cable stops on the trike frame. This is the only part I bought new.
A thumb shifter can be mounted to the stem--This can be useful if the cable is a bit short to reach to the handlebars, or if you want to keep the cable as short as practical for best shifting.
A set of cotterless cranks with a smallest chairing around 24-28 teeth. This will almost certainly have three chainrings, and you won't use the medium and large--trikes ae not stable enough for significant speed. A chainguard is useful, but the largest chainring can be turned into a chainguard by cleaning a chunk of chain, and installing it with no slack. (This assumes an even number of teeth. If the largest ring is odd, you will need a half link, avaliable at better bike shops) The original crank is too fat for a derailleur chain, but could be cleaned and re-used as a chainguard. Keep an eye on crank length--The parts bike I used was a 24", and had shorter cranks than the stock ones on the trike. This was an advantage in this case, since the original trike cranks were a bit long for my wife.
There aren't that many bikes with both 5 speed freewheels and cotterless cranks in an adult size. In my case I found a Mongoose 21 speed for crank, chain and inner cables, and a cheaper 15 speed bike for freewheel, derailleur and shifters. The cheap bike had one-piece cranks, so they were not useful. The Mongoose had a frame mounted derailleur--better, but would take more work to adapt to the trike. Kid's bikes with shifters are the most likely to have 5 speed index parts.
(Note when I say 5 speed or 7 speed, I'm talking of the rear gears. a 10 or 15 speed will have a 5 speed rear, a 12 or 18 speed a 6 speed rear, etc. )
In addition to basic hand tools, you will need a freewheel removal tool (or a shop that will remove it for you--no special tools needed to install) a chain tool and cone wrenches.