The city busses in Daegu are very cheap, and unique in that they come in three different flavours. There are the community shuttles, which are the cheapest and dirtiest, the regular busses, which are pretty much like any city bus you've ever seen and cost a fraction more than the community shuttles, and then there are the deluxe busses, which cost a bit more (but are still quite inexpensive) and have padded seats and curtained windows.
The busses are pretty regular and frequent, at least in the middle of the city. Also, several lines go as far as forty minute out of city limits, so are a good way to access tourist sites surrounding Daegu, such as temples and hiking trails.
I recommend you get yourself a transportation recharge card. It is very useful if you plan to do alot of sightseeing around Daegu. They cost W 2000 and can be bought at most subway stops. You can also recharge it at all subway stops. You can use it pay for the subway and local busses. You also get a discount rate on the subway system.