Anchorage does have a public transportation system, the 'People Mover' which works fairly well in town and is handy when the snow gets too deep to shovel your car out. When running or walking, consider the same safety precautions you would take at home. It may look like paradise, but crime does happen. Be aware that there are many fine trails that are hike/bike in the summer & cross country skiing in the winter. Many go for miles through some of the most incredible scenery in the world. Do not hike or walk along the shore of Cook Inlet, Turnagain Arm or any other shoreline without local information about safety. There are huge tide falls in the Anchorage area and it is all too easy to get trapped in a bad situation where it is almost impossible to be rescued.
In the summer, even if the sun is up and it's light outside, don't go walking/running alone in unfamiliar areas. This is also true for late nights. Your body clock may say it's time to get up and go, but if it's 4 am in Anchorage, it may not be the best time to go for a walk. Be careful of dogs that are lose. As for the bears, yes, sometimes they do come into downtown Anchorage, as do moose. These are wild creatures who have ventured into town for their own reasons. Leave them alone!! Don't end your vacation by getting hurt.