If you are looking for some added navigation on the side of your site (to further refine product choices or collections, perhaps an app is a good way to go about this? Some themes have this kind of feature built-in, it sounds like the one you've chosen does not have this kind of option.
For changing the order of menu items, you can do this the same way you would move or rearrange other items in the admin. This doc will show you how to do this with more detail. The basic premise is that you can drag and drop items into the desired order using the 6 small dots to the right of the item.
To remove the continue shopping button on your cart page, it depends on how your theme is built. Some would allow you to tweak this directly in the Theme Editor. (Online Store > Theme > click Customize > navigate to cart page > click Cart section and review options). If this is not an option then you might want to consider making some coding changes.
This tutorial shows you how to add in a continue shopping button, so removing this code should help.
Otherwise, searching the forums for your theme would be a good way to start. I usually use Google to search for the jist of what I'm after and add the words Shopify + the theme name. For example; "remove continue shopping button debut theme Shopify".
If coding is outside of your skill set, don't worry there are lots of ways that you can approach this change.
I think I'm finally getting there I've managed to create a drop-down menu and added my pages as well, I hope this looks ok for a mobile.
The only thing I can't do is remove the continue shopping button I've been into the code and removed what I thought it was but it was still there after I saved the page so I wrote it back in.
It's taken a good few hours but I'm pleased with how its looking will you take another look please and let me know what you think.