Understanding the details of thermal transport in graphdiyne and its nanostructures would help to broaden their applications. On the basis of the molecular dynamics simulations and spectrally decomposed heat current analysis, we show that the high-frequency phonons in graphdiyne can be strongly hindered in nanoribbons because of the boundary scattering. The isotropic transport in graphdiyne can be switched to anisotropic along the armchair and zigzag directions. Adding side chains onto the nanoribbon edges further reduces the thermal conductivity (TC) along both armchair and zigzag directions thanks to the reduction of heat current carried by low-frequency modes, a mechanism that arises from the phonon resonances. The uniaxial tensile strain plays a different role in the TC of graphdiyne, armchair nanoribbons, and zigzag nanoribbons. Tensile strain causes the thermal conductivities of graphdiyne, and armchair nanoribbons increase first and then get reduced, whereas for zigzag nanoribbons, the TC decreases with strain first and reaches to a plateau. The different low-frequency phonon response on strain is the main reason for the different TC behavior. For graphdiyne and armchair nanoribbons, the low-frequency heat current is enhanced gradually first and then get reduced with the increase of strain, while that of zigzag nanoribbons decreases with strain and then increases slightly. The current studies could help us understand the phonon transport in graphdiyne and its nanoribbons, which is useful for their TC engineering.