Drive, take the Greyhound, or fly into Sawyer International Airport , a former Air Force base converted in recent years to civilian use. Driving in the Upper Peninsula during winter can be treacherous and might not be safe for inexperienced winter drivers. The city is rather isolated on the shores of Lake Superior but than again so is the particular part of the state. The drive there will take you through endless forests and small lakes.
For the most part one would want to take a car, especially in winter when snow or ice covers everything. In the summer one could rent bicycles from one of the local bike shops, notably Lakeshore Bikes , which is fittingly located on Lakeshore Drive which fittingly runs alongside Lake Superior. Marquette has several bike paths , one of which runs alongside the lake from Presque Isle Park  to the intersection of M-28 and US-41. Biking on M-28 or the backroads around town is rather pleasant and the traffic isn't too harsh.
There is also county-wide bus service , which runs a few fixed routes and also offers door to door service.
Marquette has the best freeride mountain biking in the country. The trails are insane so you better gear up. In the summer Marquette is host to the Ore to Shore mountain bike race  and Superior Bike Fest, which is huge. For good riding advice, stop by Lakeshore Bikes or the Quick Stop Bike Shop. Guys at both shops are avid riders and will be happy to point you in the right direction.
As you might expect the hiking is exceptional as well, with many hikes heading up the big hills in the area that overlook the lake. One can hike up Sugarloaf Mountain or Hogback Mountain and then hike back down to the Lake for a relaxing lounge on the beach.
Check out Presque Isle Park  for a beautiful drive, bike ride, or hike right in town. It's a city park but feels much bigger. The trail around the perimeter of the island meanders through the woods, occasionally popping out for breathtaking views of cliffs and the lake. Don't miss jumping off the black rocks, accessible from a parking area on the west side of the island - a great place to hang out with friends and chat or have lunch.
Sugarloaf Mountain  is another good short hike north of town. Mostly uphill, one route takes you straight up, the other is a little gentler and includes staircases over the rougher spots. The view from the top is spectacular - a great photo spot. Also one of the best places for autumn foliage viewing.
In the winter there is a sled dog race , and the start is immediately followed by fireworks off of the dock in downtown. As well in the winter there is a cross country ski race  which goes from Ishpeming to Marquette.
In the summer there is the Hiawatha Traditional American Music Festival .
Northern Michigan University has a good hockey team and it's worth going to the games. NMU is also home to the Superior Dome, the largest wooden dome in the world (until recently) and host to NMU football and indoor track.
Uncle Ducky's Charter Fishing  has fishing trips on Lake Superior. Otherwise you can fish off of the breakwall downtown, which is accessible from the Coast Guard station off of Lakeshore Drive (just north of downtown).
The Marquette Maritime Museum  is next to the Coast Guard station. As well there is a historic lighthouse at the Coast Guard station which you can tour. You must make an appointment beforehand.
Marquette Magazine, a free monthly, always has a new list of 101 things to do in Marquette.
The city has an old-fashioned pedestrian-friendly shopping district in the downtown area. It also has a generic retail strip behind parking lots along the highway, with all the same stores you've seen in the strip of every other city in the U.S.