How do you earn legendary marks?

You can only carry 200 legendary marks at any given time per account. You can earn unlimited marks per week.

Legendary marks at first seem slow, but eventually they start piling up. Use them to infuse to ensure your light level goes up as quickly as possible - and always make sure that you identify your engrams one at at time at the cryptarch so that you can equip that gear before identifying the next one until you are at least 295 light.

The daily story offers 15 marks. It can be annoying to do if it is one of the longer missions though, especially since there is no match making.

There are often two crucible dailies offered. In destination, in the bottom left there will be a far left crucible daily which is 10 legendary marks, and the second to left option which is 15 legendary marks.

Next is the heroic strikes. These are recommended light 260 and your first three heroic strikes of the week yield 10 legendary marks. The first heroic strike of the week per character will also reward one legendary engram (https://www.bungie.net/7_Destiny-The-Taken-King/en/News/News?aid=13546). It is best to do these all in a row because of the buff for finding loot when doing sequential strikes.

The nightfall (recommended light 280) will give 15 legendary marks the first time you do it per character weekly.

Dismantling gear will yield 3-4 legendary marks depending on the gear's experience.

Many side quests will offer legendary marks as rewards, but are often only 10 or 15.