Probability of a $k$-path in a random graph
When $k$ is a constant, the threshold probability for a path of length $k$ is $p \sim \dfrac{1}{n^{1+1/k}}$. The same is true for any tree with $k$ edges. The general result is that the threshold probability for the appearance of a subgraph $H$ is $max_{J \subseteq H} n^{-|V(J)|/E(J)|}$, which happens to be the same as $n^{-|V(H)|/E|(H)|}$ for balanced graphs $H$ (such as trees, complete graphs, cycles). 

The situation is different when $k$ grows with $n$. I don't know what the threshold is, but looking at the Hamiltonian path case, I'd guess that an extra $\log n$ factor suffices.