Hero's Commendations and Protector's Commendations are acquired solely through raiding, daily missions, weekly missions, and certain achievements. They can be earned up to a cap of 250 each per week. The cap resets on Wednesday at 6AM EST / 3AM PST. Because of this cap, the "fastest" way to collect commendations is to reach the cap every week.

Summary

Before diving into the specifics, the basic gist is as follows:

?????????????????????????????????????????????????
? Activity         ? Freq.  ? Hero/Wk ? Prot/Wk ?
?????????????????????????????????????????????????
? Green Musp Raid  ? Weekly ?     150 ?      -- ?
? Red Musp Raid    ? Weekly ?      -- ?     150 ?
? Axis Raid        ? Weekly ?      -- ?     100 ?
? Shared Quests    ? Daily  ?      84 ?      42 ?
? G.L.F. Dailies   ? Daily  ?      28 ?      21 ?
? G.L.F. Weeklies  ? Weekly ?      40 ?      30 ?
? One-Shot Stories ? Daily  ?      84 ?      84 ?
?????????????????????????????????????????????????
? Total Possible   ?        ? 386/250 ? 457/250 ?
? Without Raiding  ?        ? 236/250 ? 177/250 ?
?????????????????????????????????????????????????

• If you are raid-ready:
  • Complete Green & Red Muspelheim raid for 150 of each commendation
  • Complete Axis raid for 100 Protector's Commendations to max Protector's Commendations for the week
  • Do a combination of Shared Quests and G.L.F. dailies each day for the remaining Hero's Commendations.
• If you are not raid-ready:
  • Complete the Shared Quests, G.L.F. daily mission, and all three One-Shot Stories for a total of 28 Hero's Commendations and 21 Protector's Commendations per day (196 and 147 per week, respectively)
  • Complete the G.L.F. weekly mission for 40 Hero's Commendations and 30 Protector's Commendations
  • By the end of the week, you should have a total of 236 Hero's Commendations and 177 Protector's Commendations

Raiding

Completing each of the three raids once a week will reward you with a large amount of commendations. This is the most effective way of reaching your weekly cap as you can cap your Protector's Commendations purely from raiding and almost cap your Hero's Commendations from raiding. Each of the raids split their rewards into 25 commendations per boss up to the final boss which instead gives 50 commendations.

• The Muspelheim Raid (Green) will grant up to 150 Hero's Commendations.
• The Muspelheim Raid (Red) will grant up to 150 Protector's Commendations.
• The Axis Raid (Green) will grant up to 100 Protector's Commendations.

However, raiding (particularly Musp. Red and Axis) can have gear requirements that are a bit high for someone who has yet to earn many commendations. As such, all of the following options are viable for those who are not quite raid-ready.

Shared Quests

Shared Quests reward commendations for their first completion each day. The Shared Quests and their completion status (denoted by the green phrase "Daily Bonus Available") are in the "Shared Quests" tab of the Mission Log.

The first completion of any Shared Quest for the day will grant 4 Hero's Commendations and 2 Protector's Commendations.

Genosha Liberation Front

The G.L.F. faction in Hammer Bay offers missions that reward commendations. The missions and their completion status (denoted by a green check mark) are in the "Influence Missions" tab of the Mission Log under Genosha Liberation.

The daily mission grants 4 Hero's Commendations and 3 Protector's Commendations. The weekly mission grants 40 Hero's Commendations and 30 Protector's Commendations.

One-Shot Stories

One-Shot Stories reward commendations for each completion with a bonus. The bonus is available once every 20 hours, so it's effectively a daily reward. One-Shots are also different than the other missions mentioned as you can get partial rewards for partial completion for some of them, as noted below.

It's worth noting that the tooltip for One-Shot Stories claims that the reward for completion will be a Cube Shard. This is a UI bug! The One-Shot will not grant Cube Shards at all and instead will grant commendations.

• The Bronx Zoo One-Shot will grant up to 3 Hero's Commendations and 3 Protector's Commendations, granting 1 of each per boss killed (Kraven, Lizard, and Mr. Hyde).
• The March to Axis One-Shot will grant up to 3 Hero's Commendations and 3 Protector's Commendations, granting 1 of each for successfully killing Madame Hydra before she escapes and 2 of each for killing the final boss, Red Skull.
• The Vibranium Mines One-Shot will grant 6 Hero's Commendations and 6 Protector's Commendations.

Achievements

Specific achievements can grant commendations, though it's not recommended to chase these unless you are incapable of raiding and need to push towards your weekly cap. They are mostly included here for completeness.

• The first time you level any specific hero to level 60, you will receive 1 Hero Commendation from the corresponding achievement.
• The first time you kill Doctor Doom in story mode in under 180 seconds on any specific hero, you will receive 1 Protector Commendation from the corresponding achievement.

tl;dr

• Complete each of the raids once per week (as many of the raids as you can handle),
• Complete the G.L.F. weekly quest, and
• Complete your dailies:
  • Shared Quests,
  • G.L.F. dailies, and
  • One-Shot Stories.