Hi Weezard!
No sodium bicarb! The sodium is toxic to plants at a very low concentration compared to how much you will end up with if you rais pH with it consistently. Get yourself a nice bottle of pH up.
Your rain water should still get some CalMag added. Read the chemistry link in my siggy. There's some important concepts of ionic strength and pH measurements in there.
GH will supply some, but not all, of the minerals your plants would like. I would start with your rain water and then add calmag to ~200 ppm and then your GH. Monitor to see if the low leaves get yellow Ca def blotches.
Also, you can play with STYLES of growing, and nutrient systems, until you find one that gets on well with your local water sources. I keep telling people this, that when you have weird water or a problem from past use of chemmy salty ferts or soil, that a fertilizer that contains humic acid can be quite a rewarding change!
I have to get going, but I hope that is a step in the right direction!