Badrinath is reachable by road but the region around it offers some great trekking. For Hem Kund and the Valley of Flowers start from Govindghat on the road to Badrinath. I have not this trek myself as when I was in the region late October the lodges at Ghangaria (13 km from Govindghat) were closed for the season. Ghangaria is a base to make day treks into the Valley of Flowers (national park entry fee required 600 INR for foreigners) and to the sacred Hem Kund an important pilgrimage place for Sikhs. It is forbidden to stay overnight in the Valley of the Flowers National Park so you must do it as a day hike from Ghangari. There are a few very nice day hikes that can be done from Badrinath. A hike along the opposite side of the river as the road up to Mana village is a nice walk but its better to continue up the valley to the east of Mana to reach Vasudhara Falls (about 5 km from Mana). Another nice walk is to Nilkantha base camp directly east of Badrinath. With time and camping gear it may possible to head further up the beautiful glacial valley which contains Vasudhara Falls.