Hi Mitchell,
a cursory analysis of the code suggests that you should write
VectorPlot[{1, Sin[2 t] }, {x, 0, 5}, {t, 0, 5}, VectorScale -> {.03, .03, None}, VectorPoints -> 20, VectorStyle -> {Thick, Red}, Frame -> True, Axes -> True, AxesLabel -> {"x", "t"}]
where you do the vector plot. Note that I change the
x''[t] = Sin[2 t]
to
Sin[2t]
the thing is that a single equal sign sets the left hand side to the value/function on the right hand side. Which - after solving the ODE - simply gives TRUE next time you run it. That should cause the error. The second error will be a consequence of that. I have no access to a computer at this point in time, so I might be wrong, but you might want to give it a try.
Cheers,
Marco