How do I get the date of 31 days ago?

Just subtract 31 days. For example:

LocalDate current = new LocalDate(2015, 6, 19);
LocalDate x = current.minusDays(31); // 2015-05-19

To get the current date, you could use:

LocalDate current = new LocalDate(); // Default time zone

or

LocalDate current = new LocalDate(zone); // Some specific zone

Or you may want to create your own "clock" representation which is able to give you the current Instant, in which case you'd use:

LocalDate current = clock.getCurrentInstant().toDateTime(zone).toLocalDate();

(That lets you use dependency injection to write simpler unit tests with a fake clock.)