Learn English – get introduced vs getting introduced

When first getting introduced, make sure to make direct eye contact.

The highlighted part of this sentence is an adverbial that contains the participle phrase getting introduced. In this case, the adverbial specifies when something happened. Participle phrases do not include a verb. here are some more examples:

Walking on the beach, we found lots of nice shells.
When sunbathng, apply sunscreen at regular intervals

Your second example is not a complete sentence, so it's difficult to say how it is composed grammatically. Here are two (slightly adapted) possibilities...

Getting introduced to my girlfriend's parents yesterday, I noticed that her mother's eyes looked exactly like hers. - adverbial containing participle

Getting introduced to my girlfriend's parents yesterday was very stressful. - gerund phrase, which acts as a noun and is the subject of was

get introduced is present simple. You could use it like this:

When you first get introduced, make sure to make direct eye contact.

Note that, when you use get in present simple, it requires a subject. In this example, the subject is you.

Both the original (participle-phrase) sentence and this present simple sentence are grammatically correct. The present simple version is probably easier to understand, especially for non-native speakers, but native speakers would probably feel that the participle-phrase version is more stylish.