The motto of the mongoose family is "Run and find out." Rikki demonstrates his faithfulness to that family creed during his first days in the bungalow. First he checks out the cotton wool to see if it is food; then he jumps on Teddy's shoulder and peeks down his shirt. On the first day, he occupies himself by "roaming over the house." He explores the bathroom, the writing table, and the man's cigar. He can't stay in bed with Teddy very long because he has to rush out and see what every little noise is. The next day, he explores the garden. Hearing crying, he curiously inquires, "What is the matter?"  and then asks, "Who is Nag?" When he meets Chuchundra during the night, he questions him, and then he eavesdrops on Nag and Nagaina. His curiosity pays off, for he hears them plotting to kill the man and learns about their eggs that will be hatching very soon. The next day, his curiosity as well as his intelligence spurs him to find Nagaina's eggs. The innate curiosity of his breed makes Rikki an effective protector of his human family. His agility and fighting skills would not have been much help if he hadn't overheard the two cobras plotting and if the 25 cobra eggs had been allowed to hatch. Staying true to his family motto made Rikki a formidable enemy for Nag and Nagaina.

How does Rikki show mongoose curiosity in "Rikki-tikki-tavi" during his first bungalow days?

When Rikki-tikki-tavi washes up in the bungalow’s garden, he is immediately curious rather than frightened.  We are told that a real mongoose is consumed with curiosity.  Rikki-tikki definitely demonstrates this.  As soon as he recovers from almost drowning, he begins playing around.

The motto of all the mongoose family is "Run and find out"; and Rikki-tikki was a true mongoose. He looked at the cotton-wool, decided that it was not good to eat, ran all around the table, sat up and put his fur in order, scratched himself, and jumped on the small boy's shoulder.

Rikki is not afraid of people, and likes to climb on them and sit in their laps.  His biggest concern is whether or not things are edible.  He also wants to inspect the house as soon as possible.  Almost immediately he is looking around rather than sleeping.

Teddy’s father knows that having a mongoose around can be a very good thing when you are in India.  The garden may have snakes, and the snakes may come into the house.  A mongoose is the best defense against snakes.

"I don't like that," said Teddy's mother; "he may bite the child." "He'll do no such thing," said the father. "Teddy's safer with that little beast than if he had a bloodhound to watch him. If a snake came into the nursery now --- "

Rikki-tikki proves to be a very brave mongoose.  In his first encounter with the cobras, they are almost more afraid of him than he is of them.  Rikki-tikki refuses to be backed down, and cannot be tricked.  Rikki is their enemy from then on, and he makes it his goal to remove all snakes from the garden.  It is not easy, but he succeeds in killing Nag, Karait, and Nagaina and the garden is safe for people and animals.