You can start the NXT tour by speaking to Benedict Encumberyak either near the Prifddinas or Burthorpe lodestone. The first eight listed below are for F2P players, while members are able to acquire all 16 items. NXT is obviously required to partake in this event.
Pipe Top floor of Lumbridge Castle, by the northern stairs
Monocle Top floor of the western side of Falador castle (next to the herb patch)
Toffee apple Top floor of Al Kharid Castle, on the north western side (go up flight of stairs in the big room, and take another flight just outside)
Package In front of the statue in the duel arena
Fox Pelt NW part of the top floor of Varrock Castle
Crystal Low-level RuneSpan, first island, SW of teleporting in
Surgical mask South of Ashdale lodestone, near Magda (can also speak to Gudrik in Taverley near the bank to get to Ashdale)
Toy shop South of Mudskippers point, SW of fairy ring
Scarf Beacon south of the Paterdomus
Sweets Castle Wars spectating wall, north of the portals
Foot Tuska's Back, by Wizard Chambers
Cinnamon Polypore Dungeon, east of the entrance (Fairy Ring code B-I-P)
Tea flask 2nd floor of Grand Tree, to the south
Frogburger Mazcab, just north of the world window
Raw fish-like Top of White Wolf Mountain, next to the Gnome glider
thing
Money pouch Top floor of Sorcerer's Tower
Completing all 16 you will receive:
4x small lamps
2x large lamps