Bug issue is done and goes from In Progress to Verifying status.
Feature issue is done and goes from In Progress to Testing status.
When one accepts Suggestion issue and start implementing it, one change issue's status from New to In Progress status.
When a Suggestion issue is implemented, only Pending and Closed status can go next. Why? Does not Suggestion issue need to go verified or tested?
You can configure the next allowed status for an issue based on the issue's tracker, its current status and the role of the user who performs the change. This can be configured globally in Administration -> Workflows. Make sure to check all transitions you want to allow there.
See https://www.redmine.org/projects/redmine/wiki/RedmineIssueTrackingSetup#Workflow for more information about configuring workflows in Redmine.