I’ve been reading about 1,5 books about quantum physics and I’ve also watched a few YouTube videos. In one book, I learnt that there are fields, such as the electromagnetic field, which carries forces between particles (vibrations in the field) via virtual particles. But in another book and on YouTube, they instead say that every elementary particle (quarks, photons, electrons etc.) have their own fields. But what I can’t understand is what the relation of the fields I read about that carries forces to the ones with the particles? Are these forces made up of interactions between multiple fields?
There is no clear notion of what exactly a "force" is in quantum field theory (obligatory xkcd).
Every field has particles associated with it, and we tend to associate those particles/fields with "forces" whose interactions with the rest are such that they have a major influence on their behaviour even when they are not directly involved (e.g. there are no real photons involved in the process that gives the Coulomb force in its classical limit, and no real W/Z-bosons involved in the process that turns protons into neutrons in $\beta^+$-decay).
Massless particles that couple to others in the same way the photon does always produce a classical force like electromagnetism (see this question for the technical background), while massive particles with such a coupling produce a classical force that decays exponentially (cf. Yukawa potential, and see this answer of mine for the residual nuclear force mediated by pions as an example). But even this is oversimplified, since e.g. gluons are massless yet produce no classical force at all since no free color charges can occur due to confinement.
Short answer: forces are particles (except maybe gravity). But not all particles carry a force!
Look for the Standard Model. A particle carrying a force is simply a "special" type of particle called a "force carrier". The photon is the most common example of a particle being also a force carrier (the electromagnetic interaction).
In QFT, there is an equivalence between particles and fields (or field excitations, to be more accurate). Thus, the following statements are equivalent (and equivalently approximative in their phrasing):
- Fields carry forces between particles
- Particles carry forces between fields
- Fields carry forces between fields
- Particles carry forces between particles
Now, when considering a particle vs. considering a force, there can be some subtle differences — as for example the occurrence of virtual particles (as you pointed out) in some interactions.