1. I am looking forward to my next visit there.
2. I am expecting my next visit there.
3. I am anticipating my next visit there.
(Are they all the same and grammatical? What is the part of speech of 'there'?)
1 and 3 are fine. 2 doesn't mean what you want it to mean!
"There" is simply an introductory word for this type of sentence construction. From what I've read, most other languages don't have an equivalent, and it's pretty much an idiom in English.
http://www.eslgold.com/grammar/there_is_are.html
It is not the subject of the sentence, but merely an introductory word.
http://dictionary.reference.com/browse/there
Scroll down here and read the Usage Note.
All three sentences are grammatically correct and convey a similar meaning, although there are slight differences in tone and emphasis.
1. "I am looking forward to my next visit there." - In this sentence, the speaker expresses excitement and anticipation for their upcoming visit.
2. "I am expecting my next visit there." - This sentence implies that the speaker is eagerly awaiting their visit and has certain expectations or things they are hoping to experience during their visit.
3. "I am anticipating my next visit there." - This sentence conveys a sense of eager anticipation and suggests that the speaker is mentally preparing or imagining their upcoming visit.
As for the part of speech of 'there', it is an adverb in all the given sentences. It is used to indicate a place or location where the speaker is referring to. In these sentences, 'there' is pointing to a specific location or destination where the speaker is looking forward to or expecting to visit.