A good resource for all things insulation is the Cold Climate Housing Research Youtube channel. One method I have seen would be to put R-13 fiberglass in the 2x4 wall, then a normal 6 mil vapor barrier, then use 2x2 furring strips horizontally on 2' centers to support 1 1/2 inch foam board on the inside of the wall. This would give you roughly R-20 depending on what type of foam you use, but a higher effective R value than a 2x6 wall because the foam layer prevents thermal bridging through the studs. You would need to hang 5/8 sheet rock vertically to finish the wall out. If you are doing the work yourself this would definitely be cheaper than hiring a spray foam contractor.
As far as moisture management is concerned, the rule of thumb I have heard is at least 2/3 of the total wall insulation should be on the cold side of the vapor barrier, and the remaining 1/3 inside.