"Complete" represents the final, fully-patched versions of all 3 CivIII releases and their respective Editors:
-- the original 16-Civ "Vanilla" release, patched to version 1.29
-- the first expansion-pack, PtW, version 1.27, which added multiplayer + 8 new Civs + new units
-- the second expansion, Conquests version 1.22, which added 8 new Civs (7 playable in the main game, + Austria) + new units + new buildings + 2 new terrain-types + new game-mechanics, showcased in the 9 "Conquests" scenarios (Mesopotamia, Rise of Rome, Fall of Rome, etc.)
If you play "Complete", you will actually be playing using the Conquests v.1.22 executable, with all the available content. Complete also includes multiple additional mods and scenarios, designed by people who were active right here on CivFanatics at the time. In that sense, it is the "best" version of the game.
However, Complete also includes a few bugs which were never patched. These bugs
have recently been fixed (and a whole lot more!) by CivFanatic-user
@Flintlock -- but only for the version of Complete which is available through GOG/Steam. The Steam-compatible version of Complete is also the
only downloadable version which currently allows multiplayer.
PtW does not have the Conquests-bugs, and
some people prefer playing PtW over Conquests in that respect -- but PtW also lacks the extra content, and is not compatible with most of the best (and currently maintained) CivIII mods, which were/are built to use the Conquests 1.22 executable.
Regarding 'trainers', there are various homebrews available off-site, but honestly, I don't see the point in using most/all of them, because (1) the game autosaves after every turn anyway (default number of autosave-slots = 5, but it is possible to increase this substantially, up to an entire game's worth), so an earlier save can always be reloaded if things go badly, and (2) if all a player wants is to guarantee that they win, then they can just play at the lowest difficulty level "Chieftain", which gives them
twice the productivity of the AI-controlled Civs.
That said, the utilities CRpMapSuite(?) and CivAssistII can be useful
game-assistants and are available to here on CFC: you should be able to find them by searching the "Downloads" section.
I use CAII to provide consolidated non-spoiler game-information about potential trade-deals etc. (which is also available in-game, but not in such an easily-comparable format). It runs without any problems on my WinXP laptop and Win8.1 desktop, so should also work for WinVista and Win7; however, many Win10+ users have reported difficulties getting it to run on their machines.
CRpMapSuite
is reportedly compatible with Win10, but provides less information than CAII.