Are the different quarks and leptons different kinds of oscillations of the same underlying quark field and lepton field, or same kinds of oscillations of different up, down and so on quark fields, and electron, electron neutrino and so on lepton fields? Or is there just a single field underlying all other fields?
Quantum field theory in general doesn't know about quarks and leptons. It knows about bosonic and fermionic fields (e.g., scalar fields, vector fields, spinor fields) and their interactions.
So think of quantum field theory as a framework. The Standard Model of particle physics fills this framework with specific content: specific scalar, spinor and vector fields that are believed to represent physical reality.
The specific fields in the Standard Model are:
- doublets of left-handed charged and uncharged leptons;
- right-handed charged lepton singlets;
- doublets of left-handed up- and down-type quarks;
- right-handed up-type and down-type quark singlets;
- the $SU(2)\times U(1)$ gauge fields (four vector bosons) of the electroweak interaction;
- the $SU(3)$ gauge fields (eight massless gluons) of the strong interaction; and
- the Higgs field, responsible for electroweak symmetry breaking, the masses of three of the electroweak vector bosons, and fermion masses.
Lastly, the fermions (leptons and quarks) in the theory each have three generations.
So... a whole bunch of fields arranged in highly non-trivial ways, and the fundamental field content describes the theory before symmetry breaking. But the basic idea is that essentially yes, each different type of fundamental particle is a unit excitation of its own field, i.e., the up-quark field differs from the down-quark field, which differs from the top-quark field, the electron field differs from the muon field, and so on.