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Figure 1. To prove by induction, one need prove that a statement P(n) is true for all natural numbers n. For example, suppose P(n) is the statement "n is positive". We know P(1) is true because 1 is positive. We also know that if n>0, then n+1>n>0, so if P(n) is true, then so is P(n+1). In this way, it follows that P(1), P(2), P(3), ... are all true.
Figure 2. The contrapositive is easily messed up. Taking examples for statements X and Y according to this scheme and working through the depicted equivalency, on the left, and inequivalency, on the right, may help strengthen your intuition.
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