I live in Copenhagen, my office is in Carlsberg Byen which isn't far from Valby. It's a nice enough area to wander around in the evening. Public transport is
very good, from Valby s-train station it would only take 10-15 mins to get to the city centre, so you don't necessarily need to stay around your hotel location. The absolute best way to see Copenhagen is with a bike though, you can see so much more and just jump off when you see something worth exploring more.
If you want to remain near Valby you could go for a walk through Søndermarken and Frederiksberg Have, the zoo and an old palace is also between the two. The Carlsberg Byen area next door also has the Carlsberg elephant gate/statues that tourists seem to love, and the Carlsberg brewery if you're into that. Obviously most of the sights are in the city centre though.
The posts above mentioned most of the main tourist sights to see. As expected living here, personally I think the best bits of Copenhagen are away from the tourist sights, but you have to visit them as a first timer. If you only have one or two days/evenings to explore, from Valby I'd suggest getting the s-train in to central station and doing a counter clockwise walk around the city and harbour, or get off at Norreport station and going clockwise - basically what Flaevius mentioned above. Most tourist things are pretty much inside the same area, some of them worth seeing (and grouped by closeness):
- Get off train at Norreport station
- Torvehallerne, Rundetaarn, Strøget
- Rosenborg Slot & Kongens Have
- Marmokirken, Amalienborg
- You could also walk out to Kastellet, Gefion fountain and the little mermaid here, but IMO they're not that exciting
- Ofelia Plads (mostly for being next to the harbour and the view across to the opera house), Nyhavn, Kongens Nytorv
- You could also walk across Inderhavnsbroen (bridge) to the outside street eats area for a beer and food (it's a mini Reffen)
- You could then continue on through the canals of Christianshavn, and back across Knippelsbro (bridge)
- Gammel strand, Christiansborg slot and some of those mentioned above - slotsholmen, krigsmuseet and the architecture centre
- Radhuspladsen, Tivoli area
- Get on train at central station
Or reverse to go counter clockwise.
If you want to get out of the city centre itself, of the places not mentioned, Faelledparken is nice, the five lakes and Dronning Louises Bro are nice, Vesterbro/Kødbyen, and some of the neighbours and streets in Østerbro, Nørrebro and Frederiksberg C are nice. You could see all of those with a bike. Reffen is nice for a beer and food in the sun, but it's pretty far I think if you only have an evening or two and no bike. Renting a goboat and floating around the harbour with a few beers is also nice, but probably a bit boring on your own.
A little further away, Amager Strandpark for the beach, Nordhavn, and Flaevius noted some even further away like Helsingør, Hillerød and Roskilde, and I'd add Dyrehavn (get off at Klampenborg train station) if you fancy a walk through the forest.
+1 for the Lagkagehuset and Warpigs mentions as well
