Best method for using custom PHP on homepage

Hey wordpress community!

Right, Heres the gist, I want to add custom PHP to my homepage and want to know the best method in doing so, I will be calling game statistics from a custom mysql table, it might need some calculations, Im looking for the neatest, fastest, optimized solution, Since this is my homepage I don’t want to cause any extra load on my already strained shared hosting.

Its for my pirate themed minecraft server https://www.piratemc.com, As you can see the theme is years old and I hardly use the blog function, I have a separate system running statistics with its own web interface https://stats.piratemc.com/ that I currently embed using an iframe (Yuck!), I hate iframes, and want to implement these stats properly.
Ive never created a plugin or widget for wordpress before, Ive only added snippets of PHP direct into the theme files.
I thought it might be best to use the built in query functions, also is there a way to cache results?

The homepage will have similar stats to the current system I have, Im in need of a new, cleaner more uptodate theme, and my stats system is getting an upgrade and I need to change the stats Ui anyway, so I could kill two birds with one stone and implement statistics to the homepage properly.

To add to this I will need a separate stats page for each user, if you click through https://stats.piratemc.com/ you can see the kind of profile data I have, and wish to show.