Resonant Raman spectroscopy studies are performed to access information about the intertube interactions and wall-to-wall distances in double- and triple-walled carbon nanotubes. Here, we explain how the surroundings of the nanotubes in a multiwalled system influence their radial breathing modes. Of particular interest, the innermost tubes in double- and triple-walled carbon nanotube systems are shown to be significantly shielded from environmental interactions, except for those coming from the intertube interaction with their own respective host tubes. From a comparison of the Raman results for bundled as well as individual fullerene-peapod-derived double- and triple-walled carbon nanotubes, we observe that metallic innermost tubes, when compared to their semiconducting counterparts, clearly show weaker intertube interactions. Additionally, we discuss a correlation between the wall-to-wall distances and the frequency upshifts of the radial breathing modes observed for the innermost tubes in individual double- and triple-walled carbon nanotubes. All results allow us to contemplate fundamental properties related to DWNTs and TWNTs, as for example diameter- and chirality-dependent intertube interactions. We also discuss differences in fullerene-peapod-derived and chemical vapor deposition grown double- and triple-walled systems with the focus on mechanical coupling and interference effects.