Completing the first quest rewards you with the featured unit. Completing the missions gives you the shards to raise them up, as well as some additional materials.
Additionally, the quests drops some items like fragments that are limited to a certain amount each month.
Speed Levelling Units
Selection quests are permanent content, so there's no need to really rush through it. But if you have plenty of resources, you can kind of raise units up somewhat quickly to have them ready for a selection quest.
Of course, this depends on how many resources you have saved up including skip tickets, awakening materials, NRG potions, Ovalites, and some JP/Resonance keys.
The first thing you're going to want to do is to look at your units and choose who you want to bring for a particular select quest. Ideally, it's better to have a balanced team with a couple of physical/magical damage dealers, then a support/healer, and lastly as an optional one is a tank.
However, unless you're a whale, you might not have enough shards to fully limit break them. In this case, you might want to go with the ones you can limit break to at least 3 or 4.
Last priority you might need to consider is if you have good gears/setup for them or not. Check the current farplane events to see if there are farmable gears that you can farm to help certain units that you want to bring.
Once you have decided on your units, you can start to speed level them up.
Simply put all of your units in a party, then equip all of them with at least 1 of each equipment type (weapon, armor, accessory). You can give them any random equipment for now since it doesn't really matter. This is so that you will have the flexibility of changing certain gears if you need to do so in certain parts of the selection quests.
For espers though, you'll want to equip proper ones that will help the units that you're equipping them to. This is just so that you can raise a bit of resonance when you spam quests with tickets.
Once that's done, simply go to any story quests and use the skip tickets. The mission that you choose doesn't matter that much, but it's better to choose missions that will drop items that you might need. Certain characters in your party might need certain memory jobs, or maybe materials for crafting equipment. If you don't need any specific item, then you can pretty much choose any quest.
This is a decent one to start with since it will also give exp and esper resonance, essentially saving some exp cubes for your units. Also, story skip tickets are given out much more easily than event skip tickets, but if you are in need of elemental alcrysts, then you can spam those dungeons instead.
If you've never raised any of these units beforehand, then you'll need a lot of awakening materials. Each unit has 3 different jobs, so raising 5 different units is... a lot to handle, even if you only focus on their main jobs. Chances are, you'll need to level up some of the jobs to a certain level to unlock skill/status nodes.
The best time to farm for them is during the days that give 2x drop for them. If you don't know what day a certain element gets the 2x bonus, you can click on the chambers then tap the "details" on the banner.
All Alcryst element chambers are always open every day, so you can still farm for a certain element if you really need it.
If you want to get the materials that you need fast, then you can use the materials of other elements and convert them to the ones that you need. You do this by tapping on the material and then the "item exchange shop" button, then "equivalent exchange".
The conversion is 2:1, so this pretty much means that it's ok to farm the double drop of a certain Alcryst chamber even if it's not the element that you need. It might even be better to do it that way since you might eventually need those elements later for those selection quests anyway.
There are also times where the game gives tickets to get some awakening materials that you can use. Check your list of tickets on the summoning screen to see if you have any.
You can also purchase some materials from the mog shop through most medals.
For this, you'll have to turn to the JP quest dungeon in the farplanes to get JP fast. This requires a key to do 1 run, but you can get upwards of 10,000+ JP per run IF you have JP increase items.
It will be cheaper to only unlock the essential skills of a unit. This will roughly take around 40k to 50k JP, so about 4 to 5 keys/runs.
When you're about to embark on the quest, in the party select screen, tap the "Battle Settings" button on the bottom left.
Turn on the "Use JP Acquisition UP Item" settings. Close the menu, then tap "Skip Battle" or "Embark".
On this screen, you can give some Ovalite(s) to your units. You can only put 1 Ovalite per unit, and since the JP key quest is only 1 run per key, you only need to put 1 on each unit. Putting the XL will give a unit 10,000 JP per run.
You can equip some Vision cards that give JP bonus, and also have a companion that uses the JP increase party effect. Make sure that they are someone in your friend list, otherwise the companion's party ability VC won't be applied.
These VCs have effects that increase JP gain. Job Crystal is available to purchase on the shop in the "Special Shop" banner. While the other two are handed out as part of an event. These don't actually increase the gain by that much, so new players that don't have these aren't really missing out. The important thing is to use the Ovalite.
You can do this pretty fast by going to the Affinity/Resonance key quest, but you can also raise them by using skip tickets on story quests as well.
If you want to use the key quest route, just keep in mind that you'll need to have your units perform multiple actions in order to gain a lot of Affinity/Resonance. It won't be like the JP quest where you can use skip ticket or auto run the quest.
There is a limit of how much Affinity/Resonance that you can get per run, so you only need to use skills about 5 or 6 times per unit, preferably the ones that are a range buffs. After you're done spamming buffs, you can press auto to finish off the stage.