This thread from a couple of years ago describes how to extract data used to plot the smooth components of a fitted gam model. It works, but only when there is one smooth v
In addition to Gavin Simpson's brilliant answer, there is now also an R package called itsadug which provides several functions for visualizing GAMs fit with mgcv.
Among these is plot_smooth (which according to the help 'plots the summed effects and optionally removes the random effects'). If I understand the documentation correctly this is close to Option 1 mentioned by Gavin Simpson.
There is also get_modelterm which returns a list (or optionally, a data.frame) with estimates of the selected smooth term. This seems to be equivalent to Option 2 (or the values returned from plot.gam, but without the plotting).