Quest 01
Quests are assigned to you by Characters Lynn-av in the game where, in general, completion of each is required to advance the Storyline. There is always at least one active Quest, and sometimes two.

What is Required to complete the Quest will vary enormously, ranging from several days (e.g. raising millions in Silver Silver-icon to Unlock a City) to one you can complete in a few seconds (eg. Initiate a Contract to Transport Passengers Passenger-icon).

Full Details of a Quest (left image) can be found by selecting the Info Icon Info-icon on the top-right of the Character that is offering it which will include Rewards for completing it.
Selecting the blue Show Me button, there, will guide you to the next step in completing the Quest by offering a green arrow pointing you in the right direction.

This same function can be accessed in-game by simply selecting the Character Avatar.

Info-icon Note - Quests generally have more than one Stage required to complete them, as in the "Handmade Snowstorm" example, above. Additionally, you may have already completed one of the Stages prior to receiving the Quest, in which case, you will receive instant success/completion Task-icon without ever reviewing (or knowing) which Quest it was.

Alert-icon Warning! There are a few errors in the quest system that still remain. I.e.:

Transport Passengers Passenger-icon from Wideport to Windhold

...while there is no Passenger Connection between the 2 Cities (and, another involving a City called Maraville).

If you find yourself struggling, following the green arrow Quest-arrow should lead you to the intended location/endeavor (I.e., the arrows reveal that you must, actually, transport passengers between Windhold & Giantfort, in the above).

Otherwise, Quests can be the quickest way to go up in Level early on where the Rewards are relatively substantial.

Additional Tasks

Additional Tasks are generally smaller Objectives for you to achieve. They are usually easier than Quests but can be some of the most difficult, time-consuming Time-icon, as well. Additional Tasks offer smaller Rewards, do nothing to advance the Storyline, and may be drawn from a pool of Level-appropriate Encounters Event-icon.

The Additional Tasks Icon on the bottom-left of the main game Screen is a Toggle that alternates display of the Full queue of Additional Tasks & Quests, and removal of the List while displaying a less-verbose Quests indication, as well.

Notable Examples

Event-icon "Increase your Airships to 5"

Task-icon Note: Airships Airships-icon in Reserve count toward accomplishing this Task.