More re-cap of what Cyclonus has previously posted =^_^=

You can participate in an Aht Urhgan Assault by going to the Commissions Agency in Aht Urhgan Whitegate. The Agency has no use for you until you complete Aht Urhgan Mission Two; Aht Urghan Mission Three is started and completed by speaking to Rytaal in the Commissions Agency.

You have completed Aht Urghan Mission Three and now you can participate in Assault. Ask Rytaal to give you a Military Tag.

Clerks at the Commissions Agency counters currently offer up a variety of Assault Missions in each of these locales:
Assault Missions in Leujaoam Sanctum
Assault Missions in Mamool Ja Training Grounds
Assault Missions in Lebros Cavern
Assault Missions in Periqia
Assault Missions in Ilrusi Atoll

Handy web link to Missions (incomplete info):
Assault Missions

If you join a party about to Assault your comrades will likely harbor the expectation you have previously 'attuned' yourself to the portal at the appropriate staging area (on some prior evening). These are the staging areas (each with its own runic portal):
Leujaoam Sanctum is accessed via the Azouph staging point.
Mamool Ja Training Grounds is accessed via the Mamool Ja staging point.
Lebros Cavern is accessed via the Halvung staging point.
Periqia is accessed via the Dvucca Isle staging point.
Ilrusi Atoll is accessed via the Ilrusi Atoll staging point.

To reach a single staging point/area on foot can consume an entire evening and two stacks of sneak oil and two stacks of prism powder, or it may take just fifty minutes. Once there (if you arrive alive) you attune yourself to the runic portal. Now you are ready to search out other staging points or join a party shouting in Whitegate for Assault in the area you just unlocked.

So to properly do Assault you need to:
1) complete a couple of ToAU Missions
2) complete a special ToAU Third Mission
3) get a Military Tag
4) travel to a couple of Staging Points and attune yourself to the Runic Portal at each one
5) gather an Assault party together in Aht Urghan Whitegate
6) go up to a Commissions' Clerk and sign up for a specific Assault Mission
7) go to the Chamber of Passage in Aht Urghan Whitegate and teleport to the Mission's Staging Point

Some Assaults are capped - bring the appropriate equipment.

And, as always, enjoy the game! =^_^=