Opening hours in the Faroes are longer than they used to be, but many smaller stores still close early on Saturday usually at 2PM and nearly everything is closed on Sundays.
Tórshavn is the obvious choice for shopping. But both Runavík and especially Klaksvik have some nice stores where you can buy clothes and other nick nacks.
It is very modern to wear wool and woolen clothing on the Faroe Islands. You definitely will find trendy sweaters, jackets and cheaper hats, shawls and gloves. Check out the shops "Sirri" and "Guðrun og Guðrun".There is only one proper shopping center - SMS Sølumiðstøð. Here is the largest supermarket, Miklagarður. There are different shops and a couple of chains - Burger King, Bath and Body Works, Vero Moda ect. For womens clothing Yasmin is a great shop to check out. The Shopping Center has glass art-work by artist Tróndur Patursson, who is very popular both in the Faroes and abroad.
There are a few second hand shops in Tórshavn. Circus down by the harbour sells cool clothing from Iceland - a bit pricy, but worth checking out.
Remember to ask for tax free and get 15% back when you leave Faroe Islands.
You should note that almost everything in the Faroe Islands is expensive. All consumer sales include 25% VAT sales tax but displayed prices are legally required to include this, so they are always exact. If you are from outside Scandinavia you can have some of your VAT refunded (http://www.woco.dk/compos...) when leaving the country.
The Faroese currency is the Faroese króna abbreviated kr. It is not an independent currency, but a version of the Danish krone, hence they are equal in value. The coins come in 25 currently being taken out of usage and 50 oyra one quarter and one half of a Króna, respectively, 1, 2, 5, 10, and 20 króna. Paper notes come in 50, 100, 200, 500, and 1000 króna. The exchange value on notes is equivalent to the Danish krone, and there is no service charge on exchange, as Danish notes are equally acceptable as the Faroese króna throughout the country. Before leaving Faroe Islands you should exchange the Faroese notes to either Danish notes or other foreign currency since Faroese notes are usually not known by banks outside Denmark.