What output do you need, and what inputs will you supply.
" 7 different movements " are those sequence steps? are those single acting motions or double acting (like a cylinder is extended when the valve is powered and retracted when the valve is off, or does it require 2 signals , one to extend, and 1 to retract?)
You have 5 analog inputs which can also function as digital inputs. you have 14 digital input/output pins of which 2 are used for serial, leaving 12 pins.
Automating a machine sometimes means you have to add complexity for safety reasons. I was involved with a machine for use in a commercial pottery. Had several pneumatic operated transfers. Original design have single acting valves. Cylinders moved when the E-Stop was pushed and moved back when the power was restored. Was very unsafe. If there was a problem it would only get worse because of the undesired motion. Had to replace all the valves with double acting and make the program more complex to protect product and workers. BUT - the machine could be stopped at any point and restarted with no undesired or unnecesary movements.
With automation you MUST plan for any and all emergency conditions and what can possibly happen when you pull power, and apply power. You would hate to have someone get into the machine and E-Stop it, only to then have the machine move and trap them further.
You may need to look into a 1280-1284 or 2560 based board for more I/O, but you have a very nice platform to learn with the Arduino.