To "run errands with someone" means "to accompany someone and complete joint missions, tasks, assignments, or chores." The use of the verb "run" means that the assignments are to be finished quickly, and that they require the people doing them to travel to different locations for each one.

Basically "Errand" means - a short journey undertaken in order to deliver or collect something. E.g. She has got to run an errand. She'll be back in a minute.

To 'run errands' literally means to quickly travel around town to get tasks done. However, people can say this casually without having to be in a hurry If I say, "I have to run some errands with my sister" it doesn't necessarily mean we are in a hurry. An example of an errand is to go grocery shopping

Can we have more text to understand it more clearly? It seems to me that you have chosen a book that is too advanced for you.