Octodad: Dadliest Catch proceeds in a linear format (though there is a secret level, Toilet and Trouble, that can be accessed via the Aquatic Fun Center stage). Each stage has objectives that Octodad must complete, shown on-screen, in order to clear them. If the stages involve Octodad in public places, he must not cause too much chaos in order to fit in. If the gauge on the screen fills, this will count as a loss and you'll have to restart from the last checkpoint, which would be the start of the current objective.
Aside from the objectives, there are three ties to collect in every stage (with the exception being Shark Naked, which has no ties), as well as achievements and trophies on the PlayStation 4, PlayStation Vita, Xbox One, and Steam versions of the game. This guide will provide tie locations as well as details on how to obtain the achievements and trophies, though take note that the Xbox version has more achievements than the PlayStation version has trophies, and the Steam version has more achievements than both PlayStation and Xbox versions.
If any ties or achievements/trophies have been missed, you can reload checkpoints within the stage or return to the stage after completing it as many times as you please. The game tracks the time that you take to complete a stage, as well as which tie(s) you're missing.