Learn Dutch: De wekker gaat om zes uur. - The alarm clock rings at six o'clock.

In Dutch, it's standard to use the verb gaan (to go) for devices that make an alert sound. Just like in English an alarm "goes off," in Dutch an alarm clock, phone, or doorbell simply "goes" (gaat) when it rings.

Since this lesson is about routines, how would I add "in the morning" to this sentence using the new grammar?

You can add 's morgens or 's ochtends. The sentence becomes: De wekker gaat 's morgens om zes uur. This shows that the alarm ringing at 6:00 is a regular morning routine.

Why do we use om for "at"? Could I use op or te?

For specific times on the clock, Dutch always uses the preposition om (like om zes uur). You use op for days of the week (like op maandag), so it wouldn't be correct here.

Why is uur singular here? Shouldn't it be plural since it's "six" of them?

When naming a point in time on the clock, uur always stays singular in Dutch. You say zes uur, never zes uren. The plural uren is only used when talking about a duration or a length of time, like "the movie lasted two hours" (twee uren).