any advice?

I use G96 gun treatment as a cleaner (although it does provide some lubrication as well), and Breakfree CLP as an oil. I got a Kleenbore gun cleaning kit for the caliber of my pistol -- it has the rods, brushes, and a few patches to get you started for about $10 or $15. Of course you'll need more patches with time, maybe some Q-tips, etc...

But, what I use is nothing special for CZ SP-01. I don't think there is anything special that you need. Just find what works for you. Do a YouTube search for field stripping and cleaning of your pistol and other similar ones. You'll find many, many demos of what people use and what they do.

CZ used to package their guns with a nylon brush and a little slotted plastic rod for sticking a patch on. That and an old tee shirt and a can of Breakfree CLP will work fine for the first few hundred rounds. Then you can add an old tooth brush and some Q-Tips, and maybe some Hoppes No. 9. You can used canned air, if you want, but be careful not to blow debris down into the trigger mechanism.

A tiny drop of Mobil One on the rails works well, too.