Flattop is the closest real mountain to the city and the easiest 3,000-foot summit in Alaska. The trail from Glen Alps trailhead in Chugach State Park is 1.5 miles each way; the last few hundred feet involves some easy scrambling on loose scree. From the top, on a clear day, you see Denali to the north, the Alaska Range, Cook Inlet, and Anchorage spread out below.
The trailhead is twenty minutes' drive from downtown. There is a fee to park at Glen Alps. In summer the trail is busy with locals; in winter, with snowshoes or skis, it is genuinely quiet. Avoid in poor weather — the summit is exposed and a few hikers a year get into trouble.