You start by looking in the area around the Hill Cumorah where these 'last great battles' took place.
How do we know that's the actual Cummorah and not just the place the plates where taken from.
You look near the current City of Manti, Utah where Brigham Young said ancient prophets had already dedicated the Temple site.
Moroni could have dedicated that site on his trek from Veracruz to New York.
You look in the vicinity of the grave of Zelph, that great white Lamanite warrior as pronounced by Joseph Smith
What makes us think that Zelph was from Book of Mormon times or that he or Onandagus had anything to do with Book of Mormon happenings, other than being descendant in one way or another form Lehi?
You look along the march route of Zions Camp where Joseph Smith said was located the City of Manti, the Southernmost city of the Nephite civilization.
Who says he is right?
All the prophetic declarations give you locations to check.
With the exception of Zelph, what make sus think these were prophetic?
To date, not one artifact of this civilization of millions has turned up. Not in the deserts, forests or praries. Nothing at all.
What are you looking for? A menorah? A Jesus Fish?
Joseph was commanded to send missionaries to preach to the lamanites in the borders around Missouri (think that was it, my references aren't here) so why not start where God told him lamanites actually were?
Who said that's where they alaways have been, that they didn't migrate from another place?
So far we have as many artifacts from the Nephites as we do from Harry Potter.
You are assuming that the tons of stuff that occupies museums around the world had nothing to do with Book of Mormon peoples.