Is the word moving as used here in these two sentences gerund or a present participle?
- The five stages of moving to a brand new place.
- One of the most exciting parts of moving to a new place.
Gerund are like nouns, and participles are like adjectives. You can replace moving with a noun and keep the structure of the sentence as it is:
The 5 stages of [a move] to a brand new place
One of the most exciting parts of [a move] to a new place
Only a noun-like phrase can be the object of a preposition, like of. It is therefore a gerund in both examples.