Here are two suggestions for nice day trips in the area around Tyinholmen. The peaks are easily accessible and they are also of such a character that you will experience a great view with nice nature experiences on the way:
The rust egg
Rusteggen is a characteristic reddish mountain ridge that lies above and northwest of Tyinholmen. There is easy access up the gorge from Tyinholmen - over Sukkenes bro (a large rock that lies over the stream from Sløtatjern) - further up to the path junction between Skogadalsbøen and Fondsbu/Tyinholmen and then straight north and up the ridge that ends at the cairn at 1640 m where you has a majestic view towards the Skagastøl peaks, Falketind and the Uranos massif.
Estimated time: 5 hours
Nice peak, 1540 m above sea level, in the center of Koldedalen. From Tyinholmen, the easiest way is to drive into Koldedalen and park after the 3rd bridge. Alternatively, walk from Tyinholmen (approx. 1 hour longer round trip). You aim towards a green flank to the right of the ridge. Pass a small pond and follow the stream upwards. Crossing a couple of snowdrifts. More options up the last piece. Magnificent views towards Falketind and Hurrungane in the west.
Estimated time: 3 hours
The closest of the more characteristic peaks in the area around Tyinholmen - Breikvamsnosi rises straight up on the northwest side of Tyinvannet. Here you can choose to drive in and park at the third bridge - on the road into Koldedalen. The path then enters the left side of Koldedøla - and continues towards the glacier fall on the first Koldedalsbreen before cutting up to the left up along the ridge on Breikvamsnosi. A little hilly and that there may also be some snowfields that you have to pass. The trip then goes straight up to the cairn at 1720 meters and you have a great view over Tyinvannet and eastwards towards Bygdin, Galdebergtind and Torfinnstindene.
Breikvamsnosi has been referred to in several places as one of Jotunheimen's most beautiful ski slopes and there is often snow here until well into July!
Estimated time: 4 hours