The most iconic of classic cocktails is also burdened by many misconceptions. The basics are simple - gin and vermouth, chilled and served "up". The variations are many, and mostly for the worse.
First, a Martini is made with gin. A Vodka Martini is made with vodka, and a Martini is made with gin.
There's no need to shake a Martini unless you are in a rush. Shaking is for cocktails with fruit juices, eggs, or cream. In the old days, stirring a Martini was a way to prove that you were using quality gin, while shaking it into an Arctic Sea of ice chips was a way to cover up cheap booze.
A Martini is not a Vesper, the famous James Bond libation. The Vesper is shaken because it is a literary conceit, symbolizing the violence of Bond's Cold War career. It can be stirred and be just as delicious.
Vermouth matters. Some dry vermouth is very good, some is not. All of it changes the drink, so use a good one that complements your favorite Martini gin. Dry vermouth is cheap, so buy a few bottles and see what you think, or even order it as an aperitif at a restaurant. They won't know what you mean, but it's not too hard to figure out.
Speaking of restaurant bars... You can tell a lot about a restaurant by the quality of its bar, and you can tell a lot about a bar by the vermouth that it uses. Vermouth is cheap, and it is used in small quantities, so the amount you save by using cheap vermouth is minuscule. It only matters to managers who try to save every damn penny because nobody can tell the difference. If you are reading this, then you can tell the difference; don't go to those restaurants.
And gin matters. You'll be amazed. Read about it at Gin.