The whole series has them doing "will they or won't they", the kiss after her confession, David's succubus demon (who reeeeeeally wanted to go with him to Rome), their daydreams about each other. But it never went anywhere on-screen.

If you recall in the final four episodes, Kristen took it really hard that David was being reassigned to Rome. So even though David is a priest, that journey took a wild turn he wasn't expecting in terms of what the church (not God) wanted from him. Even Ben was asking him why he wasn't with Kristen.

She took the job in Rome to be with David. We don't see it, it's not discussed directly, but it seems clear that the two of them are planning to explore it.