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The electron scattering at surfaces, interfaces, and grain boundaries is investigated using polycrystalline and single-crystal Cu thin films and nanowires. The experimental data is described by a Fuchs--Sondheimer (FS) and Mayadas--Shatzkes (MS) model that is extended to account for the large variation in the specific resistivity of different grain boundaries as well as distinct top and bottom surfaces with different scattering specularity $p$. Textured polycrystalline Cu(111) thin films with thickness $d$ $=$ 25--50 nm are deposited on a stack of 7.5-nm Ta on SiO${}_{2}$/Si(001). Subsequent annealing results in small-grain (SG) thin films with an average grain size $\overline{D}$ that increases from 90 to 120 nm with increasing $d$. Corresponding large-grain (LG) thin films with $\overline{D}$ $=$ 160--220 nm are obtained by depositing 100--200-nm-thick films, followed by an in-situ anneal and a subsequent etch to match the thickness of the SG samples. Nanowires are fabricated from the SG and LG thin films using a subtractive patterning process, yielding wire widths of 75--350 nm. Single-crystal and LG layers exhibit a 18--22$%$ and 10--15$%$ lower resistivity than SG layers, respectively. The resistivity decrease from SG to LG Cu nanowires is 7--9$%$. The thickness and grain size dependence of the resistivity of polycrystalline and single-crystal Cu layers is well described by an exact version of the existing FS $+$ MS model but is distinct from the commonly used approximation, which introduces an error that increases with decreasing layer thickness from 6.5$%$ for $d$ $=$ 50 nm to 17$%$ for $d$ $=$ 20 nm. The case of nanowires requires the FS $+$ MS model to be extended to account for variation in the grain boundary reflection coefficient $R$, which effectively increases the overall resistivity by, for example, 16$%$ for 50 \ifmmode\times\else\texttimes\fi{} 45 nm${}^{2}$ wires. The overall data from single and polycrystalline Cu layers and wires yields $R$ $=$ 0.25 \ifmmode\pm\else\textpm\fi{} 0.05, and $p$ $=$ 0 at Cu-air and Cu-Ta interfaces.