Learn Russian: Я должен съездить в аптеку за новой аптечкой. - I must drive to the pharmacy for a new first aid kit.

In Russian, when you go somewhere to fetch, retrieve, or pick something up, you must use the preposition за followed by the Instrumental case. If you used для (which takes the Genitive case), it would imply you are doing a favor for the benefit of the first aid kit, rather than going to retrieve it!

Why use the verb съездить instead of поехать?

While both involve traveling by vehicle, поехать simply means to set off or head in a direction. съездить specifically means to make a round trip: to go somewhere and come back. It is a perfective verb and is exactly the one you want for running a quick errand.

The words в аптеку and за аптечкой look very similar. Are they related?

Yes, they share the same root! аптека (seen here in the Accusative case for direction as в аптеку) means "pharmacy." The word аптечка (seen here in the Instrumental case as аптечкой) is a diminutive form that specifically means a "first aid kit" or a small home medicine box.

Could I say мне нужно съездить instead of я должен съездить?

Absolutely. Мне нужно (I need to) is very common and sounds slightly softer, focusing on a personal need. Using the short adjective я должен (I must / I am obligated to) emphasizes a stronger sense of duty or strict necessity to get the task done.