It is indeed very important to first make yourself clear that velocity is a vector (its magnitude is a scalar and called speed; English is great in being precise enough to distinguish these concepts clearly). The use of vectors is of great help in physics from day 1 on anyway. So it's good to learn about it.
One first has to start with a space-time model. In the introductory physics course you start with the (apparently) most simple successful space-time model, the Newton-Galilei space-time. According to this model, space and time are independent of any physical processes, and any observer describes space as a three-dimensional Euclidean space of points and time as an orientied one-dimensional continuum. You can visualize this space-time as a straight line (depicting the real numbers in the usual way) along which in each point there is a three-dimensional Euclidean space.
You can discribe the three-dimensional Euclidean space by arbitrarily singling out one point ("the origin"). Then each point can be addressed by a vector, i.e., an arrow pointing from the origin to that point, the position vector ##\vec{x}##. The origin itself is given by the null vector. Using three rigid rods of unit length, perpendicular to each other, giving three vectors ##\vec{e}_j##, you have a Cartesian basis, and you can describe each point by the three components of the position ##\vec{x}=x^j \vec{e}_j##, where ##x^j## are real numbers, and ##j## runs from 1 to 3. If in an equation 2 indices (one upper, one lower) are the same, you understand that one takes the sum of the corresponding three expressions:
$$\vec{x}=x^j \vec{e}_j \equiv \sum_{j=1}^3 x^j \vec{e}_j.$$
Now the most simple system is a "point particle", which is a small macroscpic body (small compared to the typical scales of its motion; e.g., in astronomy at first approximation you can consider the Sun and the Earth as point particles orbiting each other due to gravitational interaction). It's position is given as a function of time ##\vec{x}=\vec{x}(t)##.
Then the average velocity between two times ##t_1=t## and ##t_2=t+\Delta t## is defined as
$$\langle \vec{v} \rangle_{\Delta t}=\frac{\vec{x}(t+\Delta t)-\vec{x}(t)}{\Delta t}.$$
The meaning is immideately clear: The difference of two vectors in the numerator gives the arrow (displacement) of the body from its initial position ##\vec{x}(t)## to its final position ##\vec{x}(t+\Delta t)##, and dividing by the time ##\Delta t## it took to make this change of position, gives you a precise measure of how fast the particle is going and, also pretty important, from where to where it precisely went.
Now you can let ##\Delta t \rightarrow 0##. Then you get the momentaneous velocity, which tells you the direction and magnitude of the change of position during an infinitesimal time interval ##\mathrm{d} t##. This is called a derivative with respect to time, i.e.,
$$\vec{v}(t)=\lim_{\Delta t \rightarrow 0} \frac{\vec{x}(t+\Delta t)-\vec{x}(t)}{\Delta t}=\frac{\mathrm{d}}{\mathrm{d} t} \vec{x}(t)=\dot{\vec{x}}(t).$$
