Norway is a fairly safe country in general, and Tromsø is no exception. Violence is usually limited to drunk 19 year olds fighting in the taxi line at 4 in the morning. Theft is not unheard of, though, and don't leave your camera unattended.
Earlier, the local drug addicts used to beg for money along the main street. They have now started selling "Virkelig", a local version of The Big Issue, and they have now been replaced by beggars from the Balkans. Neither represent a danger, though.
Far more life threatening are outdoor activities. Tourists occasionally try unguided glacier walks, deep sea fishing, hiking and off-piste skiing without being properly trained or equipped, once in a while with fatal results. Do not try any glacier walks on your own. Deep sea fishing and off-piste need good training. Don't over-estimate yourself when hiking in the mountains, although there is a mountain for any level. Most accidents could be avoided by seeking local advice tourist information, Troms Turlag etc.
In winter, the city centre is occasionally a giant ice rink. Or bobsleigh track. To save your neck, make sure your shoe soles have plenty of rubber as opposed to plastic, and consider purchasing a pair of crampons.
contact
Free internet is found at the Public Library in the city centre ask the main desk for WiFi access. Burger King also has free WiFi. The student house Driv see Cafes section also has terminals, and unlike the library they stay open a lot longer. Coin operated machines are found at Dark Light and at Meieriet.
Free wireless zones are found in many places around town, including Peppe's Pizza and Kafé Verdensteatret, where it is free of charge. Many hotels also have it, but often charge you.