(note - this answer ignores the Tim Burton re-make)
There are four ways of viewing the PotA series (including the modern re-tellings). Each one will, essentially, track the story of a character or a theme:
Option 1: Production order [the story of Cornelius]
Option 2: Chronological-Retro [the story of the planet]
Option 3: Semi-Chronological-Limited [the story of Taylor]
Option 4: Chronological-Modern [the story of Caesar]
Option 5: Comprehensive-Destiny [you can't escape the ape] (kind of a joke answer ;)
The Story of Cornelius:
Introduces Cornelius, Zira, and his family and follows their journey through space/time in a linear / timey-wimey fashion
Planet of the Apes
Beneath the Planet of the Apes
Escape from Planet of the Apes
Conquest of the Planet of the Apes
Battle for the Planet of the Apes
Chronological-Retro
The original story of the planet in as linear a fashion as possible:
- Escape from the Planet of the Apes
- Conquest of the Planet of the Apes
- Battle for the Planet of the Apes
- Planet of the Apes
- Beneath the Planet of the Apes
Semi-Chronological-Limited:
The story of Taylor - with no history on the origin of the planet:
- Planet of the Apes
- Beneath the Planet of the Apes
Chronological-Modern
Revised history of the story of Caesar [remains to be seen if current series will directly link to the original PotA but, this list assumes that is the intention given WftPotA film]:
- Rise of the Planet of the Apes
- Dawn of the Planet of the Apes
- War for the Planet of the Apes
- The Planet of the Apes
- Beneath the Planet of the Apes
- [at this point - uncertainty if the 'Cornelius' arc should be employed]
** Comprehensive-Destiny **
In which retro and modern are combined to produce a time loop demonstrating that the planet was meant / doomed no matter what timeline shenanigans happen
Notes on Production order:
1) The primary reveal at the end of the original Planet of the Apes film is one of the most iconic and referenced endings in science fiction. To watch the films 'out of order' in relation to production is to remove the underlying mystery of the original. That being said, this is really only a consideration for somebody entirely unfamiliar with the work. Dramatic irony, of course, is always at play but there is something to be said for that first time 'ah!' moment
note on the answer - this could be more detailed with plot points probably (but would require spoiler tags) - may do so as I have time, or if it is used as a starting point, happy to convert to Community