Stockholm

Carol, it might help to know your interests but I am about your age and I took my mom (she was 73) so these were the things we liked. Stockholm's Old Town (called Gamla Stan); the Vasa museum (the Vasa is a 17th century warship that they dredged from the sea floor); Drottingholm Palace; and Millesgarden. About Millesgarden - mom and I both love Carl Milles sculptures. That's important because the place is beautiful but there's not really a tour so it helps to know a little bit about the sculptor ahead of time. Also, depending on how you go, there's a bit of a walk so if you have trouble walking, be sure to plan your access carefully. I was told (after we walked up a steep hill) that we could have gone by bus and been dropped pretty much at the doorstep if we'd come from another direction.
Lodging-wise: We like the Scandic chain. Scandic Grand Central worked well for us but if you post your budget somebody else might have a more picturesque type place (not a chain).

Margaret's suggestions are all very good. I'd also recommend the City Hall (Stadshuset) tour. This is where they hold the Nobel Prize Dinner on December 10th of each year. The tour is excellent. The Tour times for January are here. With only one or two days and given that the trip is in winter, I'd probably focus on the Vasa Museum, Stadshuset, and Gamla Stan. Millesgården and Drottningholm are much nicer in good weather, so I'd save them for a future summer trip.

You will want to be somewhere in the central part of the city. It's Stockholm, so there really isn't any particular area that is "rough", especially in the city center (the "scary" parts are all out in the suburban rail communities, and even those aren't actually scary). Take a look at this map: http://www.hotellstockholmcity.org/wp-content/uploads/2011/11/SL-Karta-Tunnelbana1.png Anything between these stops is convenient: Tekniska Högskolan to Liljeholmen Karlaplan to Hornstull Thorlidsplan and Gullmarsplan
Kungsträdgården and Stadshagen All of these are quick to get to the Central Station, which is where you will be arriving (from the Airport) and departing. As for what to see? Most things are centrally located. City hall (Stadshuset) is a masterpiece and very enjoyable. Gamla Stan is a great. Sunday at noon is the changing of the guards at the Royal Palace. Skansen is an open air museum that many people enjoy, although kids like it better than adults. The vasa musuem has a well preserved man-o-war from the 18th(?) century. Östermalms Saluhallen is a great old market and a good place for lunch. Södermalm is the trendy area where shopping and dining can be had. Check out Restaurang Pelikan and Himlen for two good places to eat. If you are interested in a true and current taste of Stockholm, Södermalm also has one of the best kebab places in the city: Amida Restaurang on Folkungagatan. Another sight to see is the view from the bluffs of Södermalm. Take the subway to Mariatorget and then walk north. You'll be rewarded with a great view of the city (and if you do it right, a couple nice shops). Hope that helps. I lived there for two years, so I all of this is pretty recent info. ;)