One way to change the styling for the homepage search bar is to modify the widget instance CSS (Not the widget itself but the instance).
Follow the below steps.
With the page open, Ctrl + Right-click on the search bar and choose "Instance in Page Editor".
This opens the page editor with the instance of the widget already selected in the horizontal tree matrix.
Scroll down the page to the CSS section and find the Input and Button styling.
Make the appropriate modifications. Maybe something similar to the below screenshot where I made changes to the border styling and added styling for border-left and border-right respectively.
Result: You can see the border is much thicker and the line separating the magnifying icon and text is gone.
To control the size of the search bar find the "div" in the CSS styling and I would probably use a percentage or maybe just a max-width to keep the responsiveness.
Baseline:
Modified:
Result:
Hopefully that helps. Just make sure you are modifying the widget instance and not the widget itself.
