More newbie stuff: What do I with bitters?
I find that bitters pair better with brown liquors, in general. Get thee some bourbon, rye, whiskey, etc. And, while Angostura is fine, I really prefer Fee's and Regan's, as they have more complexity to me.
This already may be a cocktail, but gin, sweet vermouth and bitters sounds interesting to me, but I really love all three of those ingredients.
I disagree about the pairing better with brown spirits. Gin's botanicals has a lot of overlap with those in bitters and pair well with Angostura, Orange, and others.
The 2:1 gin to sweet vermouth with Angostura is frequently called the Artillery. The better known drink adds a dash of orange bitters and is called the Hearst (named after the newspaper mogul). Although the Hearst is traditionally made as an equal parts drink (more modern is the 2:1).
If you get some benedictine, you can make a Gypsy:
2 oz vodka
1 oz Benedictine
goodly dash of Angnostura
Stir and strain into chilled cocktail glass; express the oils of a swath of lemon peel, and discard or toss into the drink.
You can also make "pink gin", which is just gin and a dash of Angostura; method of preparation and garnish should be the same as above. I haven't tried it myself, but it's recommended by Ted Haigh in "Vintage Spirits & Forgotten Cocktails".
I tend to prefer brown spirits; I can't say which type of spirits aromatic bitters go better with, but I would second invinotheresverde's suggestion to pick up some straight rye and / or other whisky. Then you can make Old Fashioneds, Manhattans (and a lot of Manhattan variations), etc.
Technically, you should use Old Tom style gin, but if you get some Maraschino liqueur, you could also make a Martinez (Sweet Vermouth, Gin, Maraschino, aromatic bitters).
I love Pink Gin. It's best when you give a heavy hand with a few solid dashes of bitters - to me the whole point is the balance of Angostura and gin.
A British friend of mine introduced me to a drink she called a Long Vodka (though I think that's just a generic term for vodka mixed-drink): A couple of dashes of Angostura on ice in a highball glass, shot of vodka, fill with Sprite. It's refreshing and kind of a poor man's Pimm's.