La Little Mermaid of Copenhagen you can reach it either on foot or by boat. It is located a little north of the old town and is only a 20-minute walk away, while every half hour starting at 10 am it can be reached by boat from Gammel Strand.
In the early 1900s the famous Little Mermaid now symbol of Denmark it was sculpted by Edward Eriksen and bought by the owner of the Carlsberg breweries. The statue is quite small and a bit insignificant but since you've come up there, why not go and see it? A tip, get the Copenhagen Card, will allow you to use all public transport, bus, train and metro and you can visit most of the museums.
Does not include the link between Copenhagen and Malmo but it also offers discounts on some services related to Sweden, a really good card. The cost is DKR 199 for 24 hours (about € 26) and DKR 429 for 72 hours (about € 57).