Allow me to make three hiking recommendations, going from easiest to most difficult.
Easiest: the Partnachklamm, a deep gorge through which the Partnach river flows. The hike is mostly flat, with only a very gentle uphill slope in some parts. The trail head is located right by the Olympic Stadium, and the entire hike should take little more than an hour.
Less easy: Ride a ski gondola to the summit of Wank (the mountain on the northeast side of town, across the valley from Zugspitze). Say hello to the cows on the summit, and follow the trail down the back side of the mountain. It winds through some gorgeous pastureland, then comes out on the front side again in the hills above Partenkirchen. Don't miss the Wallfahrtskirche Sankt Anton. The gallery outside the church is decorated with haunting photographs of local sons who died in the war. The interior of the church also has some interesting paintings that depict the various foreign armies that have marched through GaP over the ages (and the last one, the US army, has never left). The hike is mostly downhill and lasts about 3 hours. You may need a map to avoid taking some wrong turns.
Hardest: At the end of the Partnachklamm, continue uphill to Köningshaus am Schachen, Mad King Ludwig's private mountain chalet. It can only be reached by a long, strenuous hike, so it attracts a mere tiny fraction of the crowds that Neuschwanstein and Linderhof receive. Despite the considerable effort required to reach it, the chalet definitely deserves a look. The first floor has rather low-key, tasteful decorations, but the second floor bursts with color and is modeled after a Turkish harem (or at least, what people in the 19th century thought a Turkish harem looked like). It's a 6 hour round trip hike, and quite steep in places. If you're not physically fit or have mobility problems, don't even consider it. Hiking boots, trekking poles, a Camelback, layered outdoor clothing in case the weather goes bad, and a topographic trail map are recommended.
The Partnachklamm is close enough to the center of town that you can probably walk there, although if needed, you can ride the bus. You would probably want to ride the bus to the Wank gondola station.