Do particles exist according to quantum field theory? I've been reading that in QFT only fields exist and a fluctuation in these fields is what we see as particles.  So in this sense can we just totally disregard the notion of particles and just think of them as fluctuations of fields?
 A: I think this is largely a matter of terminology.
The way QFT is usually taught describes the fields in terms of Fock states, and it is the excitations of these Fock states that describe the particles. Note my terminology here - I'm not saying particles are the excitations of the Fock states, I'm saying they are described by the excitations of the Fock states. The quantum field is an operator field, i.e. a mathematical object, and it's hard to see how this mathematical object can have excited states that actually are the observable particles.
But then Steven Weinberg in his QFT books starts with the particle states rather than the field i.e. exactly the opposite approach. So it's not obvious that we can claim the field is the fundamental object and the particle states only manifestations of it.
At the end of the day we can experimentally observe the objects that we call particles, and the question of precisely how we describe them mathematically seems a rather abstract one.
A: To answer the question directly, yes. I disagree with John's answer: Fields are certainly fundamental in QFT - Weinberg starts from history and the reader's scattering intuition from QM but his volumes are definitely based on fields. In some cases it is downright incorrect to speak of particles so the distinction is important.
A particle describes the phenomenon of a quantum field excitation where the underlying mathematical structure is that of real/complex/spinor valued field over some spacetime manifold. In Weinberg, there is a discussion of little groups which I think very elegantly connects the two notions: Particle states are labeled by their little group representations. The quantum mechanical spin for instance can be thought of as little group SU(2) associated with the rotational invariance in a particular frame. So the particle state would be labeled by the momentum in the frame and the representation of SU(2) which come in half integers. This is one way one would think about energy and spin quantum numbers from a QFT perspective.
Usually it is okay to think phenomenologically - a local excitation of the Dirac field can be thought of as a particle, sure. What if I have a strongly interacting theory? What if I have collective excitations? What if I want to talk about non-local operators? The particle intuition will fail you the deeper you delve into the subtleties of QFT. One might argue that this intuition is a valuable learning tool, especially coming out of QM but you should think of it as training wheels and do away with them ASAP if you are interested in high energy theory. 
