Your best bet for happening nightlife in the city is in the Yasi-golmok area, where dozens of bars, cafes, and nightclubs are concentrated. The central shopping district also has a bunch of cafes, noraebang, hofs (beer halls), and bars. Most of the places I've listed are casual, are foreigner-friendly, and have no set dress code, but Koreans usually dress up to go out after dark. The city's gay district is near the express bus terminal.
The area around Samdeok Fire Station downtown is where most expats and foreigners living in Daegu hang out. I would mention bars and clubs there, but the turnover rate is so high that by the time you read this, the place will probably be closed. The largest concentration of foreigner-friendly bars and clubs is on Rodeo Street, which runs north-south from the fire station. Also in the area is "Hof Alley," which is lined with German-influenced beer bars.
The Kyungdae area around Kyungpook University is also a happening spot for nightlife. There are plenty of bars, nightclubs, noraebang, and DVD-bahng that cater to the college-age crowd. The area starts across the street from the university's main North gate, the bukmun. It's a 5-minute taxi ride from either the Dongdaegu or Keungogae stations off the subway.
In Dong (East) Daegu, there is also a handful of foreigner-friendly bars in Siji-dong. The streets just east of the Yeonho subway station (off of line 2) are home to a small concentration of bars catering to the foreigners who've come here to teach English in the many hagwon (privately run after-school learning academies) in the area. A couple of the bars even have weekly trivia nights and open mics.
Note: This information was accurate when it was published, but can change without notice. Please be sure to confirm all rates and details directly with the companies in question before planning your trip.