I’d suggest that the two “beyond basic” things to do are to learn to develop a wet, slick lather, and to learn how to get a very close shave with low pressure. These are two technique features that help minimize skin irritation while still creating great shaves. And these techniques give you access to using very aggressive razors comfortably; including straight razors.

I think that you are on the right track by thinking of branching out with a different soap. I will second what others have said about Stirling. However, there are artisans known for creating soaps that benefit the skin. Wholly Kaw, Zingari Man, and Barrister and Man immediately come to mind. All of these soaps compare favorably to Proraso. All come in a plastic tub, but you can buy samples at Maggard’s.