A dynamics programming approach is quick and easy. The number of ways to climb one stair is $1$. There are $2$ ways to climb two stairs: $1$,$1$ or $2$. For 3 stairs, there are $4$ ways:
($1$,$1$,$1$)
($1$,$2$)
($2$,$1$)
($3$)
For four stairs, consider what step they came from to land on the fourth stair. They could have hopped straight from the 1st, done a double from #2, or used a single step from #3. The ways to get to each of these steps are $1+2+4=7$ ways to get to step 4. The pattern can then be extended:
$4$ steps: $1+2+4=7$ ways.
$5$ steps: $2+4+7=13$ ways.
$6$ steps: $4+7+13=24$ ways.
Thus, there are $\boxed{\textbf{(E) } 24}$ ways to get to step $6.$