Can I track how much time is spent on backlog items and tasks?

Yes, you can do time tracking in two ways. You can either enable a "Used" field in the backlog items and tasks:

This allows you to enter the time spent on each backlog item and task. Or, you can use detailed time tracking:

This allows you to keep a detailed time log on each backlog item and task, and to view and export the time log for the project. This is useful if you want to track time for use in accounting or billing of customers.

To enable time tracking, open the project dialog, click the "Settings" button in the upper right corner, and select the type of time tracking you would like to use in the project:

When using detailed time tracking, you can also select which unit to use for the time tracking, in the "Unit for time tracking" setting:

You can also set the default settings for these options in new projects, in your customer account settings, if you're an administrator. Note that this only affects new projects created after your change, it does not affect already existing projects.

When you've enabled time tracking, you can register the used time in the backlog items and tasks once they have been moved into "In Progress" and beyond. There will either be a "Used" field that you can edit, or a "Used" field with a plus icon next to it that you can use to add time entries.

When you're using detailed time tracking, you can view the tracked time in the "Time" tab:

If the tab is not shown, click the "More" button at the right side of the tabs, and select the "Time tracking" checkbox: