Disclaimer: i have only tested this on the latest Steam version. there may be other things needed for the Rockstar Games Launcher version.

GTAIII v1.0 EXE [archive.org]
Important for mod compatibility and may also help if your unmodified game crashes upon launch but may also make the game ask for disc 2 (see SilentPatchIII)

Ultimate ASI Loader [github.com]
Required for loading ASI mods.

SilentPatchIII + DDraw Component [gtaforums.com]
Fixes a long list of issues in-game. Will also prevent the game asking for Disc 2 if it does ask at all.

ThirteenAG Widescreen Fix + Widescreen Frontend [github.com]
Better widescreen as well as fixing the unaligned aiming issue.

Ginput [gtaforums.com]
Adds full controller support with 5 controller presets and button prompts for Xbox and Playstation (configurable in INI file).

Limit Adjuster [github.com]
Allows changing the constraints that let the game perform on systems in it's time. Important for precisely setting max framerate to avoid timing issues that get worse with framerates above 60 (configurable in INI file, be sure you're modifying the GTAIII section and not the other games)

if you intend on uninstalling GTAIII but want to keep the mods around you can install these mods to an empty folder somewhere so you can install all the mods in one swift motion when you reinstall the game.