This is all done inside an Asset Catalog.
For the logo, open your asset catalog and select the image. Now, select the Appearances drop down on the right, and select Any, Light, Dark.
The Asset Catalog will now let you add different images for Any (pre-iOS 13), Light and Dark for that particular image:

For the background color, you'll need to define a custom Color set inside your asset catalog.
Press the + button at the bottom of your list of assets, and select New Color set.
Select the Color set that appears in your assets and, again, adjust the Appearances dropdown to read Any, Light, Dark.
You can now provide different colors for each mode using the RGB sliders underneath the Appearances dropdown:

Now, inside your storyboard you'll be able to select your custom color, and it will adapt depending on the mode running on the device:

You can test this by flipping the Interface Style widget at the bottom of the Interface Builder window between light and dark mode:
