There are lots of great places for cocktails in Ann Arbor. Drinks at all of these places averages around $10. Here are some favorites:
The Raven Club-A new establishment with an already established scene. Zack runs a great bar with a pitch-perfect bartender personality and a constantly changing drink line-up. Good live music. Food used to lag the drink, but there is a new chef... My drink recommendation: Ill-Fashion (when it is available), and Paloma (which usually is available).
327 Braun Ct-Also known by many as BAR Bar, since its only sign simply says "BAR". Smaller place with spare food options. More of a hipster scene with eclectic background music. Great drink line-up, and Luke makes everyone feel welcome with his Texas hospitality. My drink recommendation: Tequila Mockingbird (although be prepared for the heat of the tabasco).
The Grange Bar-Upstairs of the Grange. Don't let the white tablecloths fool you. This is a fun spot with fantastic food and drink. Grab the leather couch and chairs if they are free. Jen puts a lot of thought into every drink. Don't miss her (too infrequent) cocktail tasting events. My drink recommendation: GGGinger.
Zingerman's Roadhouse-The oldest and best known establishment of this list. Deserves a great deal of the credit for raising the standard of food and drink in Ann Arbor. Everyone knows the food is top notch (if a bit pricy), but the Roadhouse has (long before it was trendy) been practicing and preaching the virtue of great American Cocktails for as long as it has existed, and at prices in line with other nice bars. A few years ago, its cocktail menu spanned 8-10 pages! Recently they've pruned it down some, but it still has a couple of attractive options for every palate. I recommend two sweeter drinks: The Knickerbocker (are you man enough to order a pink drink?) and the Buster Brown.
My recommendation for all of these places: sit at the bar, chat with the bartender, and ask him/her to make something off-menu. Then expect something special.