An exercise in Munkres' Topology asks us to decide whether a given space, such as product space and the closure of a path-connected subset of a space $X$, is path-connected.
I'd like to know if there are some tricks to decide whether a space is path-connected?
Most obvious: given two arbitrary points, try to construct a path. This should work for the product problem (hint: if the factors are path connected, then this provides you with some paths to work with...)
Look for counterexamples. The topologist's sine curve and its variants are the most commonly encountered counterexamples to path connectedness. This should help you disprove the closure problem (consider the set $\{(t, \sin(1/t)): 0 < t < 1\}$).
Useful theorem: if a space is connected and locally path connected, then it is path connected. This is useful for open subsets of $\mathbb{R}^n$. Digesting the proof of this theorem will also provide some helpful insight into working with path connectedness.