Dongchundang Park
Songchondong is on the outskirts of the city with countryside and mountains just the other side of the motorway. The nearest mountain (visible from my apartment) is called Gyejoksan and apparently there is some kind of fortress at the top, so I'll definitely be hiking up there sometime!
The city centre is a little spread out. The old downtown area (Eunhangdong) is between Daejeon station and Jungangno. There is a large outdoor market near the station and then a mostly pedestrianised shopping district over the river with clothes/shoe shops, coffee shops etc. There is also a large underground shopping mall which runs from Daejeon station.
The bridge which leads into Eunhangdong looks a bit like a giant Croc shoe!
Then there is new downtown near City Hall in Dunsan-dong which is centred around the Timeworld/Galleria department store. There are less shops here, but lots of restaurants, bars and clubs. The main foreigner bars seem to be Yellow Taxi and Sponge. Then there are a couple of large department stores elsewhere. Say is in Seodaejeon and is a pretty big department store with lots of concessions including Uniqlo, Artbox and Kosney. There is a Lotte Department Store somewhere, but I've not found it yet!