Most stores open from 9 AM to noon or 1 PM and from 2 PM to 6:30 PM, although some remain open between noon and 2 PM. Shops are open from Monday to Saturday. Hotel stores have varying open hours so check at your hotel for these. Along Palm Beach hotel area the stores are open to 10 PM.
Above all, it appears that Arubans are very aware that their economy is completely dependent on tourism - so Arubans are polite to tourists, and even street vendors don't generally seek to rip off their customers though as in all traveling, don't let advice like this lull you into a sense of complacency. This may be helped by the fact that Aruba is a relatively expensive place to visit, so it tends to attract the reasonably well-off.
American dollars are accepted virtually everywhere at a decent exchange rate. If you have U.S. dollars, there is no need to change money into the local currency, the Aruban florin. The current exchange rate as of April 2014 given in shops is about 1.79 florins to the dollar. Because the island is a Dutch dependency, the Dutch currency i.e., Euros is easy to spend, and small change for purchases in dollars may be in florins. The island is actually not duty-free, but merchants respond well to competition on other islands, and duty free goods are offered by a few shops at the airport as visitors depart.
Oranjestad's waterfront features many vendors/stalls selling souvenirs. Ironically, many of these souvenirs are imported from the United States with island scenes/slogans, only to be purchased by Americans and brought back to America. Experienced Mexican and Caribbean tourists may recognize some souvenirs as bearing generic island-themed designs which are lightly customized for each major Mexican and Caribbean destination.
In Oranjestad, the Renaissance Mall contains various American and European major luxury brands of apparel and jewelry i.e. Tommy Hilfiger, Polo Ralph Lauren, Gucci at essentially the same prices as in the United States, which raises the obvious question of why they are in Aruba in the first place since most American or European shoppers wealthy enough to buy such goods would simply buy them at home anyway. The reason is that they are catering to Aruba's large numbers of South American visitors. Most luxury brands have traditionally refrained from entering South America or have entered and retreated due to the political and economic instability of the region. Aruba is a good compromise location as it has a relatively stable, functioning government and is very close to South America.
If you are not looking to go to an indoor mall, Caya GF Betico, is an open shopping area. This is a popular street in Aruba that is filled with luxury stores, souviener shops, jewelers, boutiques, shoe stores, and much more. You can find almost anything you need here on this street. It is recommended that you come to these shops during the early morning or late afternoon, as many of the stores are closed around lunch time.
You'll find strip malls and grocers at modest not easily walked distances west of downtown and elsewhere. They offer almost everything a visitor or resident might need for short stays or living in Aruba. Groceries and other supplies are all imported, so prices tend to be very high. You can catch the bus from the hotel areas to the largest supermarket, Super Food, which is about 10 mins from Eagle and Palm Beaches.
With numerous cruise ships visiting, downtown stores offer buys in jewelry, etc., typical of that in other Caribbean cruise ports, some at "duty free" prices. For cheese lovers, mild Dutch Gouda, in boxes or wheels, is a popular buy in supermarkets, though not the great bargain it used to be. Do ensure the integrity of the package seal from the maker to avoid spoilage and difficulty at customs inspection.
If you've flown to the island, the airport duty-free store offers alcohol at one liter per person tax free. This is a significant savings over purchasing alcohol at the hotels or downtown.