Calculating peak current of solenoids
Are these two options just different in terms of the power of the
  solenoid's movement (i.e. the 350mA will be considerably weaker than
  the 2A)?


It is (ampere x turns)\$^2\$ that determines solenoid pull force so if the number of turns on both solenoids is the same, the 2A solenoid will pull nearly 33 times better than the 350mA device (all other things being equal).


  If I went for 8 x 2A 24V solenoids would a 20A 24V power source be
  enough to power everything at its peak usage?


A 20A power source sounds adequate but would all eight solenoids be hitting simultaneously? A drummer has two hands and two feet meaning that maybe 4 solenoids is the maximum?