The main problem with "magnitude and direction" is the follow-up question: "What's a direction?" It's something that can uniquely be defined by a unit vector. Not very satisfactory, but no worse than, "A vector is an element of a vector space".
The physics we're talking about is formulated in Euclidean space, aka ${\mathbb R}^3$, and physical laws are presented as relationships between geometric objects that represent the symmetries of that space. There are several objects that do that, the scalar for instance. Since it is unchanged by rotations in space, its representation is trivial.
The simplest non-trivial geometric object is the vector, in fact in the form of spherical vectors, which are eigenstates of rotations, it is the fundamental representation of the rotational symmetries of $SO(3)$, the group of rotations in ${\mathbb R}^3$. That's a bit much for high school, so we're introduced to them in their Cartesian form, $(\hat x, \hat y, \hat z)$, where they have the interpretation as orthogonal unit arrows from which any and all vectors can be constructed. Foregoing that as too mathematical, we're left to describe them as things with magnitude and direction, and that does capture their essence.
Magnitude is not a difficult concept: if you have a velocity, $\vec v$, then the idea of $2\vec v$ is intuitive. Every student should have an intuitive idea of "direction", and the fact that if you rotate 180 degrees, you're now in "the opposite" direction, or if you rotate a full 360 degrees, your direction is unchanged is indeed a defining property of vectors. Moreover, it's easy to see why there are 3 independent basis vectors.
Compared with other geometric objects that represent the symmetries of space, that's no so bad. How would you describe a natural 2nd rank tensor? It has an alignment, but not a direction, it has magnitude....it can also have bulge. Moreover, there are 5 basis tensors. Not very intuitive.
Meanwhile, spinors have a magnitude and a direction...and a sign, so it's more than just a vector....but there are only 2 basis spinors? How does that work out?
So "magnitude and direction" isn't so bad.