  We introduce a set of four very high resolution cosmological simulations for
exploring $f(R)$-gravity, with $2048^3$ particles in $768\,h^{-1}\textrm{Mpc}$
and $1536\,h^{-1}\textrm{Mpc}$ simulation boxes, both for a
$|\overline{f_{R0}}| = 10^{-5}$ model and a $\Lambda$CDM comparison universe,
making the set the largest simulations of $f(R)$-gravity to date. In order to
mimic real observations, the simulations include a continuous 2D and 3D
lightcone output which is dedicated to study lensing and clustering statistics
in modified gravity. In this work, we present a detailed analysis and
resolution study for the matter power spectrum in $f(R)$-gravity over a wide
range of scales. We also analyse the angular matter power spectrum and lensing
convergence on the lightcone. In addition, we investigate the impact of
modified gravity on the halo mass function, matter and halo auto-correlation
functions, linear halo bias and the concentration-mass relation. We find that
the impact of $f(R)$-gravity is generally larger on smaller scales and
increases with decreasing redshift. Comparing our simulations to
state-of-the-art hydrodynamical simulations we confirm a degeneracy between
$f(R)$-gravity and baryonic feedback in the matter power spectrum on small
scales, but also find that scales around $k = 1\, h\, {\rm Mpc}^{-1}$ are
promising to distinguish both effects. The lensing convergence power spectrum
is increased in $f(R)$-gravity. Interestingly available numerical fits are in
good agreement overall with our simulations for both standard and modified
gravity, but tend to overestimate their relative difference on non-linear
scales by few percent. We also find that the halo bias is lower in
$f(R)$-gravity compared to general relativity, whereas halo concentrations are
increased for unscreened halos.
