Any suggestions from you other wood workers out there?
Well the design of it kind of begs for mortice and tenon joinery. At least for most of those joints. Of course mortice and tenon isn't easy. Easier with the right tools ($$$$) but still not easy.
Forstner drill bits and wood dowels would also work for most of it and would be easier than mortice and tenon while still a nice upgrade in craftsmanship over just screwing it together.
A pocket screw jig would also work and do a nice job if you happen to have one and you're not opposed to cheating a bit (i.e. taking the easy way out) on your joinery. I have the $100+ Kreg pocket jig kit and I use it all the time. Well worth the investment.
But yeah, a set of forstner bits and dowels would work and wouldn't be too expensive. For the back, you could run a dado in the top and bottom pieces with either a router table or dado blade in a table saw, then set the spindles and use flush insert pieces to fill the gaps in the dado between the spindles.