5/8" hollow ground of pretty much any major brand will be a good place to start, in stainless or carbon. I'd stick with carbon. Stainless, just had a different feel for me and it never responded as well to a good stropping.
A carbon razor should last several months between sharpening, even on a heavier beard with daily shaving.
This guy sells full kits
Browse Our Products for a Straight Razor Kit
The DVD seems kinda cheesy, but it's pretty helpful.
A strop is a strop. I used the basic strop from that kit for years, on several different razors, without a problem.
Also, don't store the razor or strop in your bathroom.