Header Section

Go to Store Admin > Online Store > Themes. Click Customize and then navigate to Mobile Header section.

Mobile header section from theme customization page

If you don't see the Mobile Header section, then go to Theme Settings -> Mobile Settings -> Use mobile header designs. Check that option and it will show the section. See more details here.

Mobile header configuration
Search box Add search button in the header
Wishlist Add a button to wishlist page
Compare List Add a button to compare page
Cart Dropdown Add cart dropdown
Customer Login Link Add a button to let customers log in
Text with link Add a custom Icon/text link

Drag and drop the content blocks to reorder the buttons as you want.

Header style Mobile settings
Padding Set header padding. If no value is given, default design value is applied.
Background color Set header background
Text color Set header text color

Menu You can use different navigation in mobile version. Select one of the menus you created. See how to create menus in Shopify.
Menu button title If you want to add some text under the menu activator button, then input it here
Menu background color Set background color of menu
Mobile open effect Choose one of the pre-defined effects to show your mobile menu