Leveling Up:
* Scratch: Learnt at Level 1
* Growl: Learnt at Level 4
* Defense Curl: Learnt at Level 7
* Sand Attack: Learnt at Level 10
* Fury Swipes: Learnt at Level 13
* Dig: Learnt at Level 16
* Slash: Learnt at Level 19
* Rollout: Learnt at Level 22
* Swift: Learnt at Level 25
* Magnitude: Learnt at Level 28
* Earthquake: Learnt at Level 31
* Drill Run: Learnt at Level 34
* Rock Tomb: Learnt at Level 37
* Swords Dance: Learnt at Level 40
TMs/HMs:
* Body Slam: TM20
* Double-Edge: TM34
* Rock Slide: TM44
* Strength: HM4
* Cut: HM1
Breeding:
* Quick Attack: From Scyther
* Leer: From Scyther, Machop, Mankey, and more
* Take Down: From Scyther, Machop, Mankey, and more
* Screech: From Scyther, Machop, Mankey, and more
* Payback: From Machop, Mankey, and more
* Rock Throw: From Geodude, Graveler, and more
* Mud-Slap: From Geodude, Graveler, and more
* Iron Head: From Machop, Mankey, and more
* Dig: From Diglett, Dugtrio, and more
* Sandstorm: From Trapinch, Vibrava, and more
* Stealth Rock: From Geodude, Graveler, and more
Note:
* This list assumes you have access to a trading partner for obtaining some of the moves via breeding.
* You may need to evolve Sandshrew into Sandslash to learn certain moves.
* The level at which certain moves are learned may vary depending on the level of the parent Pokémon you breed with.
Good luck training your Sandshrew!