The Roberts and Rosenthal paper (and the Rosenthal book) are what enabled me to clear the hurdle (or at least knock it down and keep running!).
I found it all a lot easier when I realized a lot of the continuous results were based on an implicit discretization. That is, a continuous space is partitioned into a countable (or finite) number of sets of non-zero measure. Then you can define the transition probabilities among the elements of the partition. Then when everything’s working, that should be a well-behaved discrete chain. The continous results for recurrence, reducibility, etc., can be stated as universally quantifying over possible partitions.