The setup in Lightning is easy - just add the "Add Checklist" Component to the page where you want to use/create checklists (e.g. Account, Lead, Opportunity, ...):
While in the page (e.g. Account), click on the Setup button (top right) and select "Edit Page":
In the Lightning App Builder, just drag the "Add Checklist" component to the area you want to add it to:
Drag the CheckListAdd component to the page.
If you do NOT see the component, make sure that you have MyDomain enabled.
If you have not MyDonain enabled, you can still follow the Classic Setup and add the Button as a Lightning Action in the Page Setup.
For the Custom Object, make sure that Allow Activities is selected in the Custom Object Definition.
(Otherwise you may receive the error message: Related To: Related To ID: id value of incorrect type: ... or )