1.According to the first image, you have added a Button web part in the modern page. This web part supports changing its color, but not support making the buttons larger and giving it border.
Button color is based on the site theme accent color. Go to Site settings gear -> Change the look -> Choose the site theme.
2.Follow below steps to design the page.
(1)In the home page, edit navigation as you desired.
(2)Edit home page -> Add a One column section.
(3)Add a Quick links web part in the One column section -> Add links -> From a link (Enter the link you want) -> Change custom image in the Thumbnail.
(4)In the Quick Links web part, choose Grid layout.
(5)Add a Vertical section, choose background as you want.
(6)Add a Quick links web part in the One column section -> Add links -> From a link (Enter the link you want) -> Choose List layout -> Turn off Show ikons.
Result:
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