The short answer
Sediment and iron filters go before the softener so they protect the resin; a carbon stage for chlorine also usually goes ahead.
Typical order
- Sediment filter.
- Iron or sulfur filter (wells).
- Carbon stage for chlorine (city water), if used.
- Water softener.
Why
Oxidized iron fouls resin (Penn State), and chlorine shortens resin life, so treat them first.
Combination systems
Some systems package a filter stage with the softener: Fleck 5600SXT 48,000-Grain Water Softener and Filter System, Aquasure Harmony Series AS-HS48FM 48,000-Grain Water Softener and Filter System.
Products mentioned

Softener and filter combo · Fleck
Fleck 5600SXT 48,000-Grain Water Softener and Filter System
48,000 grains · Fleck 5600 SXT

Softener and filter combo · Aquasure
Aquasure Harmony Series AS-HS48FM 48,000-Grain Water Softener and Filter System
48,000 grains · 15 GPM

Air-injection iron filter · iSpring
iSpring WCFM400K Air-Injection Iron Filter
Air-injection iron filter · 12 GPM
Questions readers ask
Should the iron filter go before the softener?
Yes, so iron doesn't foul the softener's resin.
Where does a carbon filter go with a softener?
Usually before, to protect the resin from chlorine.