A study of electron-impact excitation of highly charged ions in the helium isoelectronic sequence is reported. Nonresonant, intermediate-coupling collision strengths are computed for transitions up to and including the $n=3$ states. With the 13 fine-structure levels, i.e., $1{s}^{2}(^{1}S_{0})$, $1s2s(^{3}S_{1};^{1}S_{0})$, $1s2p(^{3}P_{0,1,2};^{1}P_{1})$, $1s3s(^{3}S_{1};^{1}S_{0})$, and $1s3p(^{3}P_{0,1,2};^{1}P_{1})$, there are 78 different inelastic transitions. Main calculations are carried out in the nine-state distorted-wave approximation including partial waves $l\ensuremath{\le}15$. Relativistic effects are taken into account in the Breit-Pauli approximation. Nine-state close-coupling calculations are also carried out for ${\mathrm{Fe}}^{24+}$ with $l\ensuremath{\le}4$ at a few energies for comparison (and for later work in paper II). Higher partial waves ($lg15$) are treated by the Coulomb-Bethe method using accurate relativistic eigenenergies and oscillator strengths for the target ions. The $Z$ dependence of the departure from $\mathrm{LS}$ coupling is investigated. Present results are in good general agreement with previous calculations where available, but some discrepancies are noted or clarified. Collision strengths are tabulated at energies from near the excitation energy of the $n=3$ complex to sufficiently high energies to enable rate coefficients to be computed. For transitions to the $n=3$ levels and for transitions where resonances are not important, the present values should yield reasonably accurate excitation rates (resonances are considered in paper II). The present data are required particularly for the analysis of observed spectra from high-temperature, high-density plasmas where the line intensities may depend on collisional redistribution among a number of excited $n=2$ and $n=3$ states.