This is the grand story of physics since the time of Maxwell up to today. Maxwell found ways to think of electric fields and magnetic fields as being aspects of one thing - the electromagnetic field. He had it easy because the characteristic energy relevant to unifying these is zero - the mass of the photon.
When physicists found that neutrons could change into protons, and protons into neutrons, measurements of energies and momenta of the particles involved indicated a characteristic mass of somewhere around 100GeV. The early theories using a "Fermi constant" fell apart, but Weingberg, Glashow and Salam, using an idea from Peter Higgs, came up with a good theory having W and Z bosons. These and the photon are aspect of one multidimensional field. EM + weak were then unified - after sufficient experimental verification, of course.
The strong force came to be seen as a matter of gluons. No one has yet unified the strong force with the electroweak. The energy range involved in this unification many times higher. Despite much research effort since the 1970s, and quarks and gluons commonly accepted as real, there's too little we know about how these unify with the leptons and electroweak interactions.
The key idea is that when something sufficiently complex vibrates, classical or quantum, the different modes of vibrations might start off seeming all the same, but they usually pair up or combine in ways leading to qualitatively different phenomena. Those are vague, mushy words. An example to illustrate: imagine two identical pendulums side by side, with a wimpy rubber band or spring connecting them. The pendulums could start off swinging with any different amplitudes or any relative phase, but wimpy spring will cause them to eventually swing together with identical amplitudes. This is the lowest energy state. Another state is for them to swing exactly opposite, same amplitudes. This state oscillates faster, and is easily perturbed into its lowest energy state + waste heat, sound, photons or however the system gets rid of excess energy.
With photons, W and Z - what is it that's doing the waving? No one knows. At some fundamental level, something in the "fabric" of spacetime is shaking about. In some way it has a zero-mass mode where it all swings together, with frequency time wavelength equal to the fundamental speed constant c. It has also higher energy modes where this is not so, but instead with waves characterized by masses of about 80 or 90 GeV. Something about these vibrations involves opposite motion, analogous to the pendulums swinging oppositely.
The old rubber sheet illustration of gravity is bogus. Spacetime does not have longitudinal vibrations (movement along the surface) at least it doesn't match up well with any theory. Vibrations and bending perpendicular to the surface (meaning along 4th or more dimensions outside our 3D) correspond to gravitational waves and gravitational fields, although missed the point. Gravity, outside of crazy places like black holes, is mostly a matter of rate-of-time. This is a hard concept to explain without getting into General Relativity and differential geometry. If someone has a stopwatch they'll find it takes one minute for a minute to pass by, no matter where they are. It's all about comparing nearby points of space - things seem to happen a bit faster for you as seen by me. Read about "gravitational redshift".
To write $\psi=A sin(2\pi f t)$ is to describe an oscillation. This tells us frequency and amplitude, but does not tell us if it's a steel bolt hanging on a string, a buoy floating in the see, or an avionics control system that's gone unstable.
To try to understand what it is that's doing the vibrating, you might as well pick up a book on new-age metaphysics. Even charge, plain old + and - of electrons and quarks, is a mystery what it really is. Some kind of pucker or topological twist of the fundamental substance? Electromagnetic fields are some kind of strain in that stuff? We can only write the equations describing overall behavior as waves.
Whatever space-time-energy-matter is made of fundamentally, little overlapping patches of space-thought-goop, quivering liquidy 3D membranes in higher dimensional space, strings or M-branes, who knows, all that physicists can do (for now) is mathematically describe the symmetries and energies and coupling constants of the vibrations.