First notes:
Because your account and character are shared across Classic and NGS (mainly storage and looks, not levels or skills), trying to juggle both game versions will cause an issue in inventory management due to NGS generally showering you with a lot of items, therefore it's best to start with Classic (available once you finish the NGS prologue and have access to the devices that let you change blocks). Note: PSO2 Classic is considered free DLC and has to be added to your library and downloaded to access.
All achievements are very much self-explanatory, but there are a few things to note. For example, in order to unlock the Phantom and Hero classes for Classic, you have to reach level 75 in two classes, which then gives you a title that completes a Coffy client order, which then unlocks said classes.
Level up
Fastest way to reach level 75 in two classes is get to the point where you can use a Subclass, then level up both by doing ARKS Mission dailies and weeklies (rewards get better the higher level your Main Class reaches, capping at around the time you unlock Super Hard difficulty).
A good strategy once your main class reaches level 75 is to not turn in ARKS Mission and Daily Client Order rewards, but to swap your Subclass into main class and then turn them in, this way the sub-level 75 Subclass gets 100% of the XP. Ideally, you'd want to be able to use your Subclass independently for the Bonus Quests that use Bonus Keys, since those give out a lot of XP, too. EXP Tickets are also great.
The same ARKS Mission. Bonus Key and Daily Client Order strategy can be used for reaching level 30 in all the other classes (since besides your main class and subclass, you have to reach level 30 in every other one excluding Phantom, Hero and Etoile.)
A lot of boss-related achievements can either be completed at the end of Exploration Quests or Story Quests - like Falz Hyunal in Episode 1's "The Beginning of The End".
For both "You Made the Difference!" and "An Extremely Enraged Mag" you have to trigger a Photon Blast, which are abilities triggered by a Level 30 or higher Mag - the 2nd achievement mentioned requiring a specific "Troy" Photon Blast to be used. In order to get a Mag, you have to complete Coffy's Client Order that gives you permission to have a Mag.
Once you have the Client Order complete, you have to feed your Mag with either unecessary items or devices that increase specific stats - like MEL Minis for MEL level, but it's recommended to feed the mag with items that level up stats which your Main Class uses (like it's best to prioritize RNG to level 200 for the best Ranger benefits, which means you either have to use downgrading devices for the uencessarily increased stats or solely use RNG Minis).
Once your Mag reaches level 30 in general, it unlocks the Photon Blast ability, which you have to assign to your subpalette in order to use it. In order to get a Troy Photon Blast, you have to go to the Excube Exchange Shop which is on the 2nd floor of the Shopping Center by the row of clerks within the red booth area, Buy any of the Troy PB Devices for 8 Excubes: Illios P (Troy Proi), Illios I (Troy Imera) or Illios N (Troy Nifta) and use one on the Mag.
Equip your mag and ideally, the best way to fill out your PB gauge is by killing enemies, which are found plenty of in Exploration Quests for example. Once the PB Gauge is filled, use the Photon Blast ability.
RNG based achievements
1. "Found the Lillipan!" - You have to find the hidden Lillipan in the Casino. There are various spots in which this Lillipan grafitti will spawn - usually in high up places that require using the balloons that have you float high in the air in order to reach them. The only way to find out is by checking all possible spawn locations and then simply interacting with the grafitti by pressing the Interact key.
2. "SUPER EXCITING!!" - After buying the Trick Darts in the Shopping Center's FUNporium (below the entrance to Franca's Cafe) for 5000 FUN (You can buy 1000 FUN tickets from the Excube Exchange Shop, 1 ticket costs 2 excubes.) OR finding Trick Darts in the Personal Shop through the Visiphone and buying one off for around 700k-1mil Meseta (or more, depending on the state of the economy). Once bought, go to your Personal Room (F9 > Personal Room or the Green Elevator in the Gate area) and use your console to place Trick Darts. Once placed, simply interact with Trick Darts until you score a triple 20. This can take a very short time or a very long time, it all depends on your luck.
3. "PSE BURST!" - Unlike the Bursts in NGS which happen often in Combat Zones, the Bursts in Classic tend to be very random. It's very much possibly placebo, but the most likely opportunity to trigger a PSE Burst is during Urgent Quests and Advance Quests that feature non-boss Areas. Another key thing is once you are notified a Burst has a chance to trigger, activate your Mag's Photon Blast, as it increases the likelihood of triggering it and try killing more and more enemies during this notification.
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"Restorer of Light" NGS achievement
You have to kill any Starless enemy excluding seasonal variants and the specific Vanford Lab Ruins boss spawn. In order to unlock these enemies, you have to reach a Battle Power rating of 3119 or greater, which is used to automatically fulfill the requirement of a main quest, thus automatically unlocking Rank 5 [Level 70] locations, this is around Chapter 6, where at some point you'll even get a Main Task titled: ""Starless", A New Enemy". The best location to spawn them in is the Rank 5 Mt. Magnus Combat Zone, by following the E and T markers and triggering as many PSE Bursts (and advancing through them) as possible. Eventually, a Starless enemy will spawn. Herzig is the family name for Starless Daiytl Sword and killing any variant (Silver, Golden, Malevolent, etc.) will count. On your way to this achievement, you'll encounter Nex Aelio in the main quest, so there's no need to bother with trying to have one spawn in the open world. Also another thing to note, the game heavily incentivizes completing main quests since they generally give out 1.5 million XP when completed, so they also are a huge help in reaching level 75 with any class in NGS.
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