Learn Dutch: Wij delen er vaak een mooie herinnering. - We often share a beautiful memory there.

In Dutch, er usually wants to be as early in the middle part of the sentence as possible. When it acts as a location ("there"), it comes immediately after the verb delen and before adverbs of time like vaak.

What is the difference between using er and daar here?

In this sentence, er is the unstressed version of daar. You use er when the location is already known and just part of the background. If you wanted to emphasize the place ("We share a beautiful memory THERE"), you would use the stressed word daar instead.

Could I move er to the very beginning of the sentence?

You cannot start a sentence with er when it simply refers to a location like "there". If you want to put the location in the first position for emphasis, you must use daar: Daar delen wij vaak een mooie herinnering.

Why is it mooie and not mooi? Doesn't the word een usually mean we drop the -e?