The most up to date information on improvements and patches tends to be available on the alphacentauri2.info wiki at Unofficial Patches. Download links are kept more up to date there as well.
It is highly recommended that you use a patched .exe, the original is filled with major exploits, and prone to crashing. A fully patched game will very rarely crash and almost all the major bugs are solved.
Recommendation: You should install Yitzi's patch if you're using SMAX and kyrub's if you're using SMAC. It is safe to install both.
The only caveat is that Yitzi's patch breaks backwards compatibility for alphax.txt and overwrites it. If you use old scenarios that require a custom alphax.txt this means that you may need to manually move the changes over. This will not be a concern for most users. Not using Yitzi's patch means that some newer scenarios and custom alphax.txt will not work for you.
Yitzi and kyrub's patches are both in active development and new releases are announced on the Chiron News Network. All published versions of the patches are available in the patches download directory. As of July 2015, the latest versions of all patches are linked from this wiki.
To use any patch, typically download the zip file and extract it into the game directory, overwriting the pre-existing files.
If you use SMAX, you're best served by Yitzi's patch, otherwise you should use kyrub's. The AI improvements won't make the AI too hard for new players, and instead aim for a reasonable level of challenge at "Librarian" difficulty. Tweaks mostly involve teaching the AI how to do not-terribly without cheating.
This page is not updated with the latest releases of these patches. scient no longer works on the patch project, but kyrub and especially yitzi are both still active with a couple of releases a month from Yitzi in 2014 and two releases from kyrub in that year.
Patches are primarily developed on the Alpha Centauri 2 forum.
Try PRACX, a modified renderer for Alpha Centauri with a number of other improvements. Included in versions of Yitzi's Unofficial Patch since version 3.4, also available separately.
Warning PRACX and/or SMAC are incompatible with Windows 10. Not yet clear.
Try the Sid Meier's Alpha Centauri#Unofficial patches latest copies of Yitzi or kyrub's patches.
Otherwise, might be related to a bug with automated artillery units (though I think Yitzi might have fixed this one sometime). Try turning off automation and see if it helps. More info here.
GOG.com recommends using Catalyst 13.4 or earlier but the game also works with 14.x or earlier.