In english, “~te ikimasu” & “~ni ikimasu” is very much the same but in Japanese, they were different. “~te ikimasu” represents “doing something and go” where “~ni ikimasu” represents “go and do something”

Example
Kudamono o (kau)katte ikimasu – I will buy fruits to your place
Kudamono o (kau)kai ni ikimasu – I go and buy fruits

~ni ikimasu is a combination of the following scenario
Go to post office – Yuubinkyoku e ikimasu
Buy stamps – Kitte o kaimasu
I go and buy stamps at the post office – Yuubinkyoku e kitte o kai ni ikimasu
Meaning : Kitte (Stamps)

Other Examples
I go and play at friend’s house – Tomodachi no uchi e (asobu)asobi ni ikimasu
Who would you go and meet – Dare ni (au)ai ni ikimasu ka
Meaning : Tomodachi (Friend), Asobu (Play / Visit), Dare (Who), Au (Meet)

Well, in English we normally don’t use “go and” but just “Who would you meet”. Hope this is not too confusing.