You're quite right that the other fundamental forces of Nature possess mediator particles, e.g. the photon for the electromagnetic force. For gravity, a graviton particle has been postulated, and is included in the
five standard string theories which are candidates for quantum gravity. From a quantum field theory perspective, the graviton arises as an excitation of the gravitational field. String theory, of course, postulates it arises in the spectrum of a closed string.
Mass certainly gives rise to a gravitational field, but many other quantities do as well, according to the field equations of general relativity. As you're a high school student, I'll present them as,
$$\underbrace{G_{\mu\nu}}_{\text{geometry}}\sim \underbrace{T_{\mu\nu}}_{\text{matter}}$$
Spacetime geometry, and hence the gravitational effects, are equated to the matter present in a system, which may include energy, pressure and other quantities other than mass.
From a general relativity standpoint, the gravitational field may be viewed, or interpreted, as the curvature of spacetime, which is a manifold, i.e. surface. If we take space to be infinitely large, then the gravitational field must extend indefinitely; otherwise where would we choose to truncate? Even from a Newtonian perspective, we see that given the equation,
$$F_g \sim \frac{1}{r^2}$$
gravity must extend infinitely, as we never reach the point $r=\infty$ where it is truly zero.
As you asked, if the graviton is postulated, what is the need for a field? Well, we know that particle number is not conserved; we can have virtual particle and anti-particle pair production, and as such the idea that a field propagates throughout space, and the particles are excitations of the field, is a more compatible viewpoint. In addition, the concept of a field arises because of locality. From empirical evidence we know gravitation and electromagnetism do not act instantaneously, at every point.
