
unterwegs
on the way
On the way to a destination; currently between the starting point and the destination.
Ich rufe dich später zurück, ich bin gerade unterwegs.
I'll call you back later; I'm on the way right now.
on the way / out and about
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on the way
On the way to a destination; currently between the starting point and the destination.
Ich rufe dich später zurück, ich bin gerade unterwegs.
I'll call you back later; I'm on the way right now.

out and about
Generally traveling a lot or being away from home; leading an active life that is not tied to one place.
Als Beraterin ist sie fast die ganze Woche über für ihre Firma unterwegs.
As a consultant, she is out and about for her company almost the whole week.
'Unterwegs' is made up of the preposition 'unter' (under / among) and 'Wegs', an old genitive form of 'Weg' (way). So it literally means something like 'of the way' or 'on the way'.
Ich kaufe das Brot, es liegt direkt auf dem Weg.
I'll buy the bread; it's right on the way.
The adverb 'unterwegs' is almost always used with a form of the verb 'sein' (to be). The typical construction is: 'jemand ist unterwegs' (someone is on the way / out and about).
Wann kommst du? Ich bin schon unterwegs.
When are you coming? I'm already on the way.
The exact meaning depends on the context. Words like 'gerade' (right now) or 'jetzt' (now) emphasize that someone is on a specific route. Adverbs like 'viel' (a lot), 'oft' (often), or the addition 'beruflich' (for work) tend to describe a traveling lifestyle.
Ich kann gerade nicht telefonieren, ich bin unterwegs zum Bahnhof.
I can't talk on the phone right now; I'm on the way to the train station.
away / out
Both words can indicate that someone is not in their usual place. „Unterwegs“ emphasizes that the person is on the way to a destination, while „auswärts“ merely highlights being somewhere else or doing something away from home. „Unterwegs“ fits a journey, while „auswärts“ is suitable for staying somewhere else.

Ich bin noch unterwegs und komme erst am Abend nach Hause.
I’m still on the way and won’t get home until the evening.
Ich bin heute auswärts und arbeite im Büro unserer anderen Stadt.
I’m away today and working at the office in our other city.