In long-range percolation on $\mathbb{Z}^d$, points $x$ and $y$ are connected by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$, where $α>0$ is fixed and $β\geq 0$ is a parameter. As $d$ and $α$ vary, the model is conjectured to exhibit eight qualitatively different second-order critical behaviours, with a transition between mean-field and low-dimensional regimes when $d=\min\{6,3α\}$, a transition between long- and short-range regimes at a crossover value $α_c(d)$, and with various logarithmic corrections at the boundaries between these regimes. This is the first of a series of three papers developing a rigorous theory of the model's critical behavior in five of these eight regimes, including all long-range (LR) and high-dimensional (HD) regimes. In this paper, we introduce our non-perturbative real-space renormalization group method and apply this method to analyze the HD regime $d>\min\{6,3α\}$. In particular, we compute the tail of the cluster volume and establish the superprocess scaling limits of the model, which transition between super-Levy and super-Brownian behavior when $α=2$. All our results hold unconditionally for $d> 3α$, without any perturbative assumptions on the model; beyond this regime, when $d> 6$ and $α\geq d/3$, they hold under the assumption that appropriate two-point function estimates hold as provided for spread-out models by the lace expansion. Our results on scaling limits also hold (with possible slowly-varying corrections to scaling) in the critical-dimensional regime with $d=3α<6$ subject to a marginal-triviality condition we call the hydrodynamic condition; this condition is verified in the third paper in this series, in which we also compute the precise logarithmic corrections to mean-field scaling when $d=3α<6$.