I am not familiar with the theme you’re using, but that ought to be irrelevant anyhow.
Pages can be assigned template files, the php of which defines what and how content is displayed on said page.
You do not need to modify the page templates supplied with the theme, instead create your own in addition to the ones existing already (for reference read the linked codex page). Assign that to the page set as blog page.
Whether you decide to run three separate loops or fetch the posts in one query sorted by categories (and echo markup accordingly from within the loop) is pretty much up to you. As for how to go about said loop(s) see the codex on the WP_Query class and its usage.
Whether you enqueue a separate style sheet for it or edit the themes style.css file is again up to you.
Should the original theme author ship regular updates and you be planning on making use of these in the future, you might want to decide not to modify it at all, but to familiarize yourself with child themes instead.