You may want to consider becoming a Physicist, especially if the ambiguity of engineering classes bother you. :)
Electrons and Holes Need Holes or Electrons to Flow to!
Excess electrons or holes do not cause current to flow because those excess electrons (or holes) need someplace to go to. If the electrons/holes don't "see" any places to go to, you don't get current. You need a path for the electrons and holes to flow. (This path also has to be energetically allowable, that is: the electrons need to have enough energy to follow that path. That's why batteries don't randomly spark and arc.)
Batteries Use Entropy to Make a Potential Difference
It just so happens that many batteries use entropy to keep the holes at one end and the electrons at the other. This trick is complicated, and something you should ask the Chemistry Stack Exchange, or at least in a different question here. Entropy keeps the holes and electrons apart, despite the fact that they would electrically come together.
Additionally, batteries also work by allowing ions (not electrons or holes) to flow from one end of the battery to the other. This allows the electrons to flow through the rest of the circuit instead of having them discharge like capacitors. Once again, this is where chemistry and physics meet, and you should try the chemistry stack exchange for more information.
Since electrons/holes need holes/electrons to flow to, you're safe touching one end of a battery, but in trouble if you hold both ends. If you're holding only one end of the battery, the holes/electrons don't "see" anywhere to go, so they don't go. If you connect the circuit, the holes and electrons "see" each other at opposite ends of the circuits and then flow through it to meet up. You could also get current if you brought something with a lot of holes or electrons near the appropriate end, although it would likely be in the form of a sudden static discharge instead of the smooth flow from a battery.
Maxwell's Equations, Magnets, and Induced Currents
Finally, if you take a look at Maxwell's equations, you'll find an equation with a term that relies on a change in magnetic fields to produce current. More specifically, it's those last two which explain the relations between magnetic fields and electric fields. Since you're starting your career as an electrical engineer, you may want to wait until you have a good grasp of vector calculus before tackling those.