We study level-set percolation of the Gaussian free field on the infinite $d$-regular tree for fixed $d\geq 3$. Denoting by $h_\star$ the critical value, we obtain the following results: for $h>h_\star$ we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level $h$; for $h<h_\star$ we prove that the number of vertices connected over distance $k$ above level $h$ to a fixed vertex grows exponentially in $k$ with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level $h$, at least away from the critical value $h_\star$. Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value $h_\star$ and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [AC2].