What's the best way for inexperienced cook to finely cut cabbage? I really like home-made pickled/fermented cabbage (russian style; which is bigger slice size than typical Sauerkraut; I would say 1mm thick). But I'm not a great cutter - I am very slow and find it quite difficult to do good even fine cut of a whole head of cabbage.

What is the best (two criteria for "best": fastest and/or lowest-learning-curve) way to finely cut cabbage in those amounts (1-2 medium sized heads) for someone without devoting 10,000 hours of training to get the professional chef techniques?

I'm OK with using tools (mandoline? grater?) as long as a reasonable quality tool that's recommended can be had for $20-$30 range - e.g. "Buy $200 cuisinart with special blade" isn't quite in scope here.
You can try using a good long sharp knife and a large wooden cutting board like Michael mentioned but it can be slow if you're inexperienced. Other cutting boards will dull your knife quickly which is why I mentioned a wooden one.  

The only other way I know is to use a mandoline. You'd have to first cut the cabbage small enough to use but I know for myself, it would be a lot faster than using a knife. (And I'm an experienced cook too!)

If you do much food preparation and cooking, a mandoline is very useful to have. I'm not sure if you'd consider $50 (rough estimate as I don't know where you live) expensive. The best, in my opinion, for a home cook is a Börner mandoline. It's German made and does a great job! I'm not advertising their product - only stating that I think it's one of the best. You can check this video that compares a number of different reasonably priced mandolines. 
https://youtu.be/I15-htHJF_I