I have attached a picture of the cube in the question.
An ant moves along the edges of the cube always starting at $A$ and never repeating an edge. This defines a trail of edges. For example, $ABFE$ and $ABCDAE$ are trails, but $ABCB$ is not a trail. The number of edges in a trail is known as its length.
At each vertex, the ant must proceed along one of the edges that has not yet been traced, if there is one. If there is a choice of untraced edges, the following probabilities for taking each of them apply.
If only one edge at a vertex has been traced and that edge is vertical, then the probability of the ant taking each horizontal edge is $\frac12$.
If only one edge at a vertex has been traced and that edge is horizontal, then the probability of the ant taking the vertical edge is $\frac23$ and the probability of the ant taking the horizontal edge is $\frac13$.
If no edge at a vertex has been traced, then the probability of the ant taking the vertical edge is $\frac23$ and the probability of the ant taking each of the horizontal edges is $\frac16$.
In your solutions to the following problems use exact fractions not decimals.
a) If the ant moves from $A$ to $D$, what is the probability it will then move to $H$? If the ant moves from $A$ to $E$, what is the probability it will then move to $H$?
My answer:
$A$ to $D$ then to $H = \dfrac23$
$A$ to $E$ then to $H = \dfrac12$
b) What is the probability the ant takes the trail $ABCG$?
My answer:
Multiply the probabilities: $$\frac16\times\frac13\times\frac23 = \frac1{27}$$
c) Find two trails of length $3$ from $A$ to $G$ that have probabilities of being traced by the ant that are different to each other and to the probability for the trail $ABCG$.
My answer:
$$\begin{align} ABFG&=\frac16\times\frac23\times\frac12=\frac1{18}\\[5pt] AEHG&=\frac23\times\frac12\times\frac13=\frac19 \end{align}$$
d) What is the probability that the ant will trace a trail of length $3$ from $A$ to $G$?
I don't know how to do d). Do I just multiply every single probability?
Also, could you please check to see if I have done the a) to c) correctly? I am not completely sure if this is the correct application of the multiplicative principle.
Your answers for a) to c) are correct (except for your loose use of the equals sign).
For d), note that any path of length $3$ to G will contain exactly two horizontal steps and one vertical step, the vertical step can come at any of the three steps, the probabilities are fully determined by when the vertical step comes, and all trails are mutually exclusive events. You've already determined the probabilities for the three types of trail, so now you just need to count how many of each there are and add up the probabilities multiplied by those multiplicities.