Jyväskylä is great to explore by bike. The http://visitjyvaskyla.fi website has a list of bike-hire operators, including Polkupyörä Tori.
Local buses all leave from Vapaudenkatu, near the tourist office. Tickets cost €3 to €8.20 depending on distance. Day tickets (€8 to €14) can be bought at the Linkki desk located in the same building as the tourist office – good value if you’ll be making three or more journeys.