Based on my experience with the new Tot's System, I don't think that this new system is going to properly teach new players how to use certain mechanics. A lot of players begin playing with Burning events, where they'll quickly reach higher levels, meaning that they'll be learning content such as Enhancement options and Cosmetic options when they should be focused on learning the core gameplay mechanics, such as skills, questing, and basic interface interaction. My suggestion is to rearrange the distribution of information based on their relevance and difficulty to new players.
Pets should be introduced early, maybe around level 30, as they are very simple to understand and will allow new players to earn resources such as items and mesos, as well as complete quests faster. This also teaches players a little bit about how to use the inventory system, and gives them a tangible and exciting reward for doing so.
Safety Charms and Respawn Tokens can be introduced at around the same level as Pets, as they give an item to the player that gives them a safety blanket while they are still learning the core mechanics as the difficulty increases, while remaining easy to understand and non-intrusive. Since this is being introduced at the same time as Pets, this should just be combined into one Tot's Quest.
Cosmetics should be introduced around level 100, when a player has a better idea if they want to continue playing their class after seeing their 4th job advancement.
In the community, it's generally told that players shouldn't worry about Enhancing Equipment until at least level 140, since that's when they first get their mid-game gear. Use that level range at the end of their Burning levels to teach players the next step, Enhancing their gear, by giving them direct access to their 140 Pensalir gear, as well as the information and tools regarding Enhancements that's already provided.
Familiars and Teleport Rocks should be introduced around level 60, when the player cannot rely as heavily on auto-teleports from Lightbulb Quests, and need alternate methods of getting to places. Familiars, likewise, are only really useful after this point in the game, and should only really be taught when they're going to have a noticeable impact to the player. My suggestion is to give the player a familiar around their level, instead of a Pig, since that will make the system appear stronger and more enticing to the player. Getting a weak familiar to start off with will achieve the opposite effect and make the player think that the system is weak and not worth their time.
As for the gear boxes, they should be distributed at their respective levels, being 30, 60, 100, and 140. This will help players at all of their major early job advances, and even out the amount of information they have to take in at a given point in time, making the game more accessible to new players.