There's only 3 ways to get axp at this time.
1) ICC Shuttleport - all new characters start here, and can gain 2 or 3 levels worth of axp before leaving for the mainland. You cannnot return to here, so if you didn't start here, this is not an option
2) The new alien playfields - Released in the most recent patch, these areas are for lvl 180+ and are raid oriented (20+ players in each area).
3) Aliens spawned at org cities - Any guilds that have build a city have the ability to spawn aliens at will. These spawns consist of several waves of aliens attacking on the ground, followed by a boss-type general. If these waves & general are successfully defeated, several guild members will be invited (semi-randomly) to board the alien ship that arrived to bring the aliens & general.
I'd say your best bet is going to be #3. If you're not in a guild, visit the forums, and check the Life on Atlantean/Rimor threads for any orgs that are recruiting, or ask around among your in-game acquaintances for recommednations. You'll want to make certain that the org you join has alien battles for the level range that your character is (aliens spawn based on the level of the guild members present, so having a tight level range of players is usually best). Another option to joining an org is to befriend some org members, and ask to tag along at their battles. Non-guilded players don't affect the number or level of aliens, so having extra attackers can help with success rate. The real axp gain happens on board the ship, though, so if you remain unguilded, or fight mostly at other org cities, you'll be missing out on this possibility.
Hopefully this info is helpful. IF you have any more questions, post 'em here, or if you're on Rk1, feel free to send me a /tell in-game