The following are the top 5 fishing spots within 30 minutes of Columbia, SC.  What is great about all of these, is they very accessible to the city.  The other very unusual trait about these, are they host fish which are not native to this area.  We have trout and small mouth bass.
1. Lake Murray- This is a huge 50,000 acre lake.  It hosts huge fish of all variety and is very clean.  You hear people catching 40 + lb catfish often.  You also have Striper fishing.  Go catch some monsters.  Fishing tournaments are often held throughout the year. 
2. Saluda River- This is a river where the water is taken from the very cold bottom of Lake Murray and often stocked with Rainbow and Brown trout.  The water consistently stays cold and trout can be caught year round.  Thanks to the DNR for stocking the trout.  You can catch the trout on a variety of lures or bait.  Throw a rooster tail or use kernal corn.  The river has other fish as well, but trout are the main attraction.
3. Broad River- My personal favorite.  It is the least accessible of the 5 spots.  However, it boasts tons of Smallmouth bass.  Most people fish the river for this reason.  People catch catfish, largemouth, and other panfish year round.  It is restricted to smaller boats, I take the canoe.  It is slow flowing river usually and very shallow.  There are also a lot of shoals.  Very serine river to fish.
4. Congaree River- A very popular river for huge Striper fishing.  This river forms in Columbia at  the confluence of the Broad and Saluda rivers.  There are tons of tributaries coming from inside the swamps.  You may encounter an alligator in this territory.
5. Wateree River- This river flows southeast of Columbia.  It is the furthest river away from the city but can be accessed about 30 minutes away on a main road.  Plenty of fish await to be caught and a few rapids may be encountered along the way.  This river merges with the Congaree.

The picture taken is from the Broad River.