If you want to go exploring further afield, your best option is the rail network. You’ll reach Stuttgart via Singen in just under three hours, Munich via Friedrichshafen and Ulm is a four-hour train journey away.
A trip to Zurich takes about one and a half hours. The train to Zurich departs from the Schweizer Bahnhof (Swiss station), which abuts Konstanz main station. Please note that the Swiss rail services are more expensive than the Deutsche Bahn services. A second class trip to Zurich costs CHF 33.
If you’re looking for a cheaper alternative, you might want to check out the Mein Fernbus/Flixbus long-distance coach services departing from Konstanz Allmansdorf, Döbeleplatz and Mainau Island. The two and a half-hour trip to Stuttgart costs about € 10, the three and a half-hour trip to Munich costs € 11. An € 8 ticket will get you to Zurich.