  We investigate states of rapidly rotating Rayleigh-B\'enard convection in a
cylindrical cell over a range of Rayleigh number $3\times10^5\leq Ra \leq
5\times10^{9}$ and Ekman number $10^{-6} \leq Ek \leq 10^{-4}$ for Prandtl
number $Pr = 0.8$ and aspect ratios $1/5 \leq \Gamma \leq 5$ using direct
numerical simulations. We characterize, for perfectly insulating sidewall
boundary conditions, the first transition to convection via wall mode
instability and the nonlinear growth and instability of the resulting wall mode
states including a secondary transition to time dependence. We show how the
radial structure of the vertical velocity $u_z$ and the temperature $T$ is
captured well by the linear eigenfunctions of the wall mode instability where
the radial width of $u_z$ is $\delta_{u_z} \sim Ek^{1/3} r/H$ whereas $\delta_T
\sim e^{-k r}$ ($k$ is the wavenumber of an laterally infinite wall mode
state). The disparity in spatial scales for $Ek = 10^{-6}$ means that the heat
transport is dominated by the radial structure of $u_z$ since $T$ varies slowly
over the radial scale $\delta_{u_z}$. We further describe how the transition to
a state of bulk convection is influenced by the presence of the wall mode
states. We use temporal and spatial scales as measures of the local state of
convection and the Nusselt number $Nu$ as representative of global transport.
Our results elucidate the evolution of the wall state of rotating convection
and confirm that wall modes are strongly linked with the boundary zonal flow
(BZF) being the robust remnant of nonlinear wall mode states. We also show how
the heat transport ($Nu$) contributions of wall modes and bulk modes are
related and discuss approaches to disentangling their relative contributions.
