You should converge each calculation with respect to all the parameters; however, some parameters are relatively transferable between related systems. Let's consider the two main ones in a condensed-phase simulation:
Size of basis set
Brillouin zone sampling ("k-points") (Only relevant for periodic systems)
The basis set is largely dependent on the external potential (usually meaning the nuclear Coulomb or effective core/pseudopotential). Increasing the size of the basis set means describing shorter-wavelength variations of the wavefunction, and the wavefunction typically has the shortest oscillations near nuclei, where the nuclear potential is strong and changes sharply. However, the nuclear potential is so strong in these regions that the wavefunctions tend to be fairly insensitive to the chemical environment the atoms are in, so this is quite a transferable property between any simulations using the same nuclear potentials. (Note that this isn't quite the case if you also have localised bond-centred basis states, as that obviously depends on the bonding.)
In contrast to basis sets, the quality of k-point sampling you need depends on the nature of the bands in the system, so is very sensitive to changes in chemistry, bonding, defects etc. k-point sampling is generally not transferable between systems, even introducing a single vacancy can change the k-point sampling you need considerably. Since the k-points sample the Brillouin zone, the size of which depends on the real-space simulation cell, the appropriate quantity to consider is the k-point spacing (or, equivalently, k-point sampling density), rather than the number of k-points directly. You need smaller k-point spacing for systems with degenerate states at the Fermi-level (e.g. metals), in order to sample the Fermi surface well. The k-point spacing you need also interacts with any Fermi-level broadening you are using; the greater the Fermi-level broadening, the smoother the Fermi surface and so you can use a larger k-point spacing (i.e. fewer k-points).
For PAW and ultrasoft pseudopotentials there is often an auxiliary grid used to represent the hard "augmentation" charge, and this also should be converged. This is very similar to the basis set size, in that it is a property of the pseudopotential and transfers well to other systems with the same pseudopotentials.
Finally, beware of relying on total energy as your sole property of interest when converging. Different chemical and materials properties can have very different convergence rates. Typically forces and stresses are more sensitive than total energies, and vibrational properties are even more sensitive, as are NMR chemical shifts. When converging with respect to basis set size, k-points etc. you should always focus on the property you're actually interested in or a reasonable proxy.