I don't know how to solve your original problem, but I can describe how to solve a simpler version of it. Given numbers $a_1,\dots,a_n$ and a target $y$, if you only allow operations +,-, you can count how many operations it will take to reach $y$.
In particular, this amounts to asking for integers $x_1,\dots,x_n$ such that
$$a_1 x_1 + \dots + a_n x_n = y,$$
while minimizing $|x_1| + \dots + |x_n|$.
This can be solved using integer linear programming. Introduce variables $w_1,\dots,w_n$ and the constraints $w_i \ge 0$ and $-w_i \le x_i \le w_i$ as well as
$$a_1 x_1 + \dots + a_n x_n = y,$$
and then minimize the objective function $w_1 + \dots + w_n$.
While integer linear programming is NP-hard in the worst case, when the numbers aren't too large, I expect that this will yield an efficient solution.
Introducing the *,/ operators makes your problem much harder.
To deal with your full problem, you might consider using the A* algorithm. For instance, you might build a heuristic function that estimates the distance to the target by using integer linear programming to get an overestimate of the distance to the target (you know that the distance using +,-,*,/ operators will be at most the distance using +,- operators, and the latter you can compute efficiently using ILP). I don't know for sure whether this will work well or not, but it's something you could try.
