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

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.

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 $b$.

Let $C$ be on the line through $B$ that is perpendicular to the segment $AB$ with the distance $BC$ equal to $1/2.$ Take any half-line $L$, not through $B$, that originates at $A$ and intersects the segment $BC.$ For some $n\in \Bbb N$ there is a path along $L,$ starting at $A,$ determined by $n$ points $A=A_1,...,A_n$ where $A_j,A_{j+1}$ are distance $1$ apart for each $j<n,$ and such that the distance from $A_n$ to $B$ is less than $1.$

Let the point $D$ be such that $A_nD=BD=1$ nd $D\not \in \{A_1,...,A_n\}.$ Then the path determined by $\{A_1,...,A_n\}\cup \{D\}$ is a path of the desired type.

The cardinal of the set all such $L$ is $2^{\aleph_0}$ so there at least this many paths of the desired type, joining pairs of points .

And each path is determined by a function from some $\{1,2,...,m\}\subset \Bbb N$ into $\Bbb R^2.$ The set of all such functions has cardinal $2^{\aleph_0}$ so there are at most $2^{\aleph_0}$ paths of the desired type.