Online story/missions and Hero rank
Hero Rank
To begin with your maximum number of tickets are very low and so is your team cost. It’s very important to level up your hero rank and aim for level 120, where you will gain the max team cost of 206. At this point you will also have 103 max tickets, which will make it easier to participate in events that require constant use of your tickets.
The first missions of the online story (chapter 1, 2 and 3) all offer very low amount of experience. However, chapter 4 gives the best exp/ticket ratio out of any missions in the game (excluding some limited-time events). I would highly recommend that you get through chapters 1, 2, 3 and 4 as early as you can. Once you reach the shadow temple in chapter 4 you unlock the best mission for exp. The final battle here gives you 2440 exp for 9 tickets.
After reaching this mission I recommend you only do one online story mission per day. Each day the newest story mission will give you double exp. You can make the most of this by doing it then waiting a day for the next mission to become double exp. There is no reason to rush through the online story. Chapter 6 and onwards become very challenging and you will need a good team to do the battles.
If you want to see the full breakdown of exp requirements and the benefits from hero rank you can see them in KD’s Hero Rank guide.
Trainer's Dojo
This has a few battles that teach you about some of the mechanics in the monsters you collect on the islands. I personally found this a great thing to do as the first bit of online content while going through the islands. These training missions cost no tickets to enter but reward you with 60 exp and 1.5 training points. They’re incredibly useful for getting yourself training points when you’re in need and you can grind them for exp to increase your hero rank (the last stand section, first battle is the best for this). Grinding this battle is an effective way to level up to begin with and will help you rush through the start of the online content, if you’re willing to spend the time replaying it.
Side Quests
These can be found in the “Story Events” and are much like the online story with a gem every few battles. They give more exp than the online story to begin with but ultimately become less efficient. I recommend you delay playing these missions until you have reached chapter 4 in the online story, unless you get stuck before that point.
Other Stories (e.g. The First Ones, Divine Visitors, etc.)
These are story events that have come into the game as limited-time content and have been re-released in this section. Like all limited-time events, the early battles in these are easy to finish and can be a good place for you to grab some gems and other rewards. To fully complete them you will need to have a strong team but they are a great place to pick up rare ingredients which you need to evolve legendaries.