Vodka cocktails are kind of pointless really… since it has no flavor and is essentially alcohol and water, you can mix it with anything.
Gin on the other hand, has a huge number of great cocktails in which it is the main spirit.
Things like the Pegu Club, Singapore Sling, Tom Collins, Corpse Reviver, Negroni, Gimlet, etc…
For simplicity’s sake, I’d say to start with the Tom Collins and Gimlet, and see where you go from there.
1 1/2 oz gin
1 oz lemon juice (fresh, not realemon)
3/4 oz simple syrup
Shake the first 3 ingredients, and put in a collins glass (tall glass, not a pint glass) with ice, Top off with soda water. (it’s essentially a sparkling lemonade with gin)
2 oz gin
1 oz Rose’s Lime Juice (or other lime cordial)
Shake together, and top with soda water if you want.
Finally, here’s a great site for cocktail recipes: http://www.cocktaildb.com