Lanayru Tower is in East Hyrule, north of Kakariko Village. This page has advice for finding and climbing the tower.
Location of Lanayru Tower
Before you search for Lanayru Tower, it can be helpful to activate Dueling Peaks Tower first, then go to Kakariko Village. Read Seek Out Impa if you haven't been to Kakariko Village yet.
Then, if you want to put a pin in Lanayru Tower with the Sheikah Scope, you can warp to Great Plateau Tower and look northeast. You can see several towers in that direction. Lanayru Tower is the one on the right. If you pin it, the pin will be close to Kakariko Village.
Going to Lanayru Tower
From Kakariko Village, go northeast, and you will reach a fork in the path where the right side slopes upward, and the left side goes straight ahead. Go left at that fork and continue going northeast.
You will reach another fork in the road. On the map, this area is labeled Lanayru Road - West Gate. Go north at this fork, and soon you will see some smoke in the distance. Go toward it and you will find Eryck sitting by the camp fire. You can talk to him to learn more about Zora's Domain and the Lanayru Tower.
Go north from Eryck and you will soon see the Lanayru Tower. If you go straight toward the tower, you will soon find the Rutala River blocking your way. However, you can float down to an island that sticks out from the opposite side of the river, so do that. Go northwest along this island and you will pass by a sleeping Hinox. If you wake it up, you will have to fight it (or run away). It can pull up trees and use them as weapons, so be careful.
Northeast of the Hinox, there is a wooden bridge. Go across it, then go east on the bridge. If you haven't been here before, Tona will talk to you as you go across. From here, you can make your way up the slope to the Lanayru Tower. You can try to climb up the north side of this hill, but depending on how far you are in the game, it might be constantly rainy on that part of the hill.
If you make your way up the grassy slopes on the south side instead of trying to climb the rocks, you will encounter Bokoblins, Lizalfos, and Moblins on your way up. If you have arrows, you can shoot at the red Xs on the wooden posts that hold up the wooden platforms that some of the monsters are standing on. They can't swim, so they will drown if you drop them in water.
Climbing Lanayru Tower
The tower is easy to climb, so just go from platform to platform until you reach the top.