Which word is for any going-out event, i.e. temporarily leaving where one lives for some personal business in another place? "outgoing" as noun means personal expense. Thanks.
An errand is a short trip to accomplish some bit of personal business, either for yourself or on behalf of someone else. An errand is short - several minutes up to a couple of hours. You would generally not use this word for an overnight trip. Sometimes the word applies to the items of personal business and is used in the plural, but the idea that you've traveled a short distance to accomplish those things is still included. Examples:
Can you spare some time to run a quick errand for me?
She's going to the post office on an errand for her mother.
I'm taking a longer lunch break than usual because I need to run a few errands - drop off my dry cleaning, buy some cat food, and if there's time, I'll take the car through the car wash.
As @StephenS mentioned in the comments, you don't use errand for scheduled activities, just for things you do during available time. Going to school or work are things you do on a schedule, so these would not be called errands.
To be honest, I'm not aware of a single word for a trip to work or school - people just say that they are going to work or going to school.
There are words that are used for a brief trip, whether scheduled or not: outing is common; sortie, junket, and excursion are rarer. But all of these have a connotation of something done for entertainment, rather than work or school.