Draw line in runtime

For drawing lines, check out the line renderer, here's some tuts: tut1, tut2.

You should have your points already in an array or a list. Draw the lines you want between them, and have your object move through the points one by one, use a Lerping function to smooth your movement. Something like:

 IEnumerator MoveTo (Transform objectToMove, Vector3 targetPosition, float timeToTake)
 {
     float t = 0;
     Vector3 originalPosition = objectToMove.position;
     while (t < 1)
     {
         // putting the increment of t before the lerp, will make it so that by the end of the loop our object would reach 
         // its target position with 100% because t is clampped between 0 to 1 in the lerp.
         // but if we but the increment after it - in the last run - t would become something like 0.99942 or something
         // we'd lerp to that - and then t would go beyond 1 and the loop would exit - and we'd end up with our object sitting at 0.99942
         // of the road - not 1
         t += Time.deltaTime / timeToTake;
         objectToMove.position = Vector3.Lerp (originalPosition, targetPosition, t);
         yield return null;
     }
 }

Here's a very quick and dirty implementation (attach this to the object you're moving):

 using System.Collections;
 using System.Collections.Generic;
 using UnityEngine;

 public class MoveToPoints : MonoBehaviour
 {

   // Inspector variables, pre-set by you
   public List<Transform> locations;
   public GameObject trail;           // instead of drawing lines, I'm gonna instantiate a trail object along the way
   public float timeToTake = 1f;

   private Transform _transform;
   private bool hasReachedTarget;

   // cached the transform component
   void Awake()
   {
     _transform = transform;
   }

   IEnumerator Start()
   {
     foreach(Transform t in locations)
     {
         StartCoroutine(MoveTo(_transform, t.position, timeToTake));
         while (!hasReachedTarget)
             yield return null;
     }
   }

   IEnumerator MoveTo(Transform objectToMove, Vector3 targetPosition, float timeToTake)
   {
     float t = 0;
     Vector3 originalPosition = objectToMove.position;
     hasReachedTarget = false;
     while (t < 1) {
         t += Time.deltaTime / timeToTake;
         objectToMove.position = Vector3.Lerp(originalPosition, targetPosition, t);
         Instantiate(trail, _transform.position, Quaternion.identity);
         yield return null;
     }
     hasReachedTarget = true;
   }
 }

Result:

Credits to the original MoveTo function goes to @whydoidoit from UnityGems (Specifically the Coroutines++ tutorial)