Is the set of paths between any two points moving only in units on the plane countable or uncountable?

Solution 1:

Yes, this looks convincing. For the missing step, go directly from A towards P in unit steps until the distance left is less than 2. Then use the remaining distance as the base of an isosceles triangle with unit legs, which you make point away from L.

Solution 2:

Go one unit from $a$ at an angle of $t$ to $c$.
Go in unit steps along ca until one is less than a unit away from $b$ to a point $p$.
If $p \neq b$, then draw a triangle with base $pb$ and sides of unit length adding the sides as the final steps.

As for each $t$ in $[0,2\pi)$, I've constructed a different accepted zigzaging from $a$ to $b$, there are uncountably many ways of so staggering from $a$ to $b$.