Three possible orders:
The fictional order of fictional events is:
1) Triplanetary billions of years in the past to centuries in the future.
2) First Lensman soon after Triplanetary
3) Galactic Patrol Centuries after First Lensman.
4) Gray Lensman instantaneous sequel to Galactic Patrol.
5) Second Stage Lensman instantaneous sequel to Gray Lensman.
?) Masters of the Vortex Some say it happens after Children of the Lens but most say that it happens between Second Stage Lensman and Children of the Lens which is my opinion.
6) Children of the Lens About a generation after Second Stage Lensman.
However, some people might want to read them in an order that avoids spoilers.
That order should be
1) Galactic Patrol centuries after First Lensman.
2) Gray Lensman instantaneous sequel to Galactic Patrol.
3) Second Stage Lensman instantaneous sequel to Gray Lensman.
4) Masters of the Vortex.
5) Triplanetary.
6) First Lensman. Think of Triplanetary and First Lensman as secret history lessons taught by the Arisians to the Children of the Lens as they grow up.
7) Children of the Lens.
Or a compromise order:
1) Triplanetary Omitting the first sections with the backstory.
2) First Lensman Soon after Triplanetary
3) Galactic Patrol Centuries after First Lensman.
4) Gray Lensman instantaneous sequel to Galactic Patrol.
5) Second Stage Lensman instantaneous sequel to Gray Lensman.
?) Masters of the Vortex Some say it happens after Children of the Lens but most say that it happens between Second Stage Lensman and Children of the Lens which is my opinion.
5a) Then read the first sections of Triplanetary with the backstory that the Arisians would tell the children of the Lens as they grew up.
6) Children of the Lens About a generation after Second Stage Lensman.
And I don't know the fictional order of:
Also worth noting - There are THREE authorized post- E.E.Doc Smith books that are worthy of reading by David A. Kyle : The lensman from Rigel, The Dragon Lensman and Z-Lensman. They just work the stories for these three non-human lensmen without distracting from the incredible work of E.E. "Doc" Smith.
But it would be really good to have them listed in their proper fictional order as well as the books by E.E. Smith.