Hi ewen,
As you haven't purchased components yet, maybe I can help simplify things!
There are high-current versions of the '565 which mean you won't need to use other transistors etc. I would suggest: TPIC6C595
As only one solenoid will be activated at once, the chip should cope with the required current:
Continuous drain current, each output, all outputs on, IDn, TA = 25°C... 350 mA
Peak drain current, single output, TA = 25°C (see Note 3)... 1.1 A
The chip's outputs are protected against reverse current spikes, so you won't need those diodes either.
Paul