Learn Hardcore Arabic (Modern Standard): أنا هنا اليوم مع صديق.

Arabic often allows dropping the subject pronoun because the verb usually shows who the subject is. But here there is no verb (it’s a nominal sentence), so أنا (I) is commonly used to make the subject explicit.
You can omit it in some contexts (especially in conversation) if it’s already obvious who is speaking, but أنا هنا اليوم مع صديق is the straightforward, clear form.

Where is the verb “am”? How does Arabic say “I am here”?

In Modern Standard Arabic, the present-tense verb “to be” is usually not stated. So أنا هنا literally reads I here, which means I am here.
If you need past or future, Arabic uses forms of كان:

كنتُ هنا = I was here
سأكون هنا = I will be here

What is the basic sentence structure here?

It’s a nominal sentence (جملة اسمية):

أنا = subject (مبتدأ)
هنا = predicate (خبر) meaning here
Then اليوم (today) and مع صديق (with a friend) add extra information (time and accompaniment).