How to change the look of second article at front page
One approach to do this is to use multiple views. One that displays only one node and a second one that displays the others, with an offset of one. Then it's easy to configure those views to use different view modes then. You can do that using Page Manager/Panels, Views attachments or possibly also core blocks.

One thing that can be a challenge to figure out is paging, depends on what you expect to see on the second page exactly. If you have paging, then a different approach might be necessary, for example by changing the view mode of the first view through code, through a preprocess hook or so.