Always use Noilly Prat vermouth. Accept no substitutes (well, perhaps in this hipster era of ours there are acceptable substitutes out there, but guilty until proven innocent, I say). I'm pretty sure we got Noilly Prat at Trader Joe's, but your Aptos liquor store is bound to have it, as it's the standard bearer, not niche.
I favor stirred martinis. If you don't have a stirrer, you can use a large pint glass. Put enough ice cubes in it that they will be higher than the gin. Stir quickly to chill the glass (you don't want the ice to melt, and if any water collects on the bottom of the glass throw it out before adding the gin). Add the gin (just shy of 1/4 of a cup per person) and no more than a Noilly Prat cap full of vermouth, stir rigorously for a few seconds and pour into martini glasses. The idea is to ensure as little dilution of the drink, so no extra time in contact with the ice. Take your lemon peel piece, give it a quick twist above the glass to release the aromas and drop it in. You're all set.
My martini is based on the stirrer I use, so the amount may vary but it looks like about 1/4 cup. No need to fill up the vermouth cap entirely either, it's really just meant to be a hint. Some folks even rinse their martini glasses with vermouth first then chuck it out before adding the chilled gin.
Some people also store the gin in the freezer or place martini glasses in the freezer for 30 minutes before making the drink. I sometimes do the former, but never the latter, as I don't like drinking from a wet glass and having precipitation all over my hands.