In QFT it's very common to hear (read) about the self-interactions of a field. e.g. there's the self-interaction terms of the Higgs field, that come with $\lambda^3$ and $\lambda^4$, right? But I still don't understand what it means? In my mind fields only interact with things around it, like gravity with mass, or electromagnetic field with charges. Picturing either of these last two self-interacting doesn't make much sense.
Could anyone help with the notion of the self-interaction of a field?
I'm afraid you're searching for a picture that isn't there, because you think of both fields and charges still classically. In quantum field theory, there are no fields in which localized particles move and interact with them - all the "particles" arise from the fields themselves, and all that is dynamical in this theory are the fields. However, you might think of classical field theory for a start, e.g. used for describing the oscillation of a vibrating string. Introducing "interaction terms" into that theory makes the resonant frequencies shift and the description of the oscillation, or of the propagation fo a wavefron ton the string, much more complicated.
We call a (scalar) field $\phi$ with a Lagrangian density $$ L = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi + \frac{1}{2}m\phi^2$$ "free" and fields with any other Lagrangian density "interacting". Typically, this means that the Lagrangian density is of the form $$ L = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi + \frac{1}{2}m\phi^2 + V(\phi)$$ where the potential $V$ is a polynomial in $\phi$, e.g. $V(\phi) = \phi^4$. This contains no picture of "how" the field interacts, just as the $\frac{1}{2}m\phi^2$ term contains no explanation of "how" this is the mass.
However, once we have developed perturbative QFT so far as to read rules for Feynman diagrams off from the Lagrangian, the meaning of the interaction terms becomes more clear: A term of the form $\phi^n$ allows for the appearance of vertices at which $n$ lines associated to the field meet. If we adopt the sloppy language of not distinguishing virtual "particles" from actual particles, then it means that it's allowed that $n$ particle have a localized interaction in such a diagram. The field "interacts" with itself in particular in the sense that a $\phi^4$ term allows one particle associated to the field to "split" into three of the same particles (neglecting kinematic obstructions), a process completely impossible for any free field. Without interaction terms, none of the fundamental particles could ever turn into each other, or, indeed, even couple to each other.
Conversely, the diagram language makes it clear why the "free" field is free: In the absence of interaction terms, the only thing you can draw is straight lines, never intersecting, never meeting - nothing is happening.
However, Feynman diagrams are ultimately calculational tools and not a depiction of actual "processes". In the end, there is no "picture" to the self-interaction of a quantum field because we don't have a true picture of the quantum field to begin with - our intuiton works much better with "particles", which we can barely define in the free case and which become ill-defined in the interacting case.
This is easier to understand classically. At the classical level, if you have a Lagrangian quadratic in the fields, then the equation of motion is linear in the fields, since the Euler-Lagrange equations differentiate once. This means that solutions to the equations of motion obey the superposition principle, which means that two field configurations (e.g. wavepackets) heading towards each other just pass through each other.
Higher-order terms in the Lagrangian lead to nonlinear terms in the equations of motion, so wavepackets don't just pass through each other: they can scatter. That's an interaction.
