Best way to get around is by bike. There is nextbike (bike rental) in Augsburg. Also you can use the tram. Tickets range between 1,35€ up to 13,00€. Normal price is 2,70 €. There are not many ticket inspections so you could try fare-beating.
To get further away you can hitchhike. http://hitchwiki.org/de/Augsburg
If you are willing to pay some money there is https://www.airbnb.de/
And there is the Grandhotel Cosmopolitan: http://grandhotel-cosmopolis.org/de/hotel/uebernachten/ Witch is a very cool project to support.
Meet fellow travellers on hospitality exchange networks: Trustroots
There is free WLAN on to big places in Augsburg: Königsplatz and Rathausplatz. Also in the Information Office and in some trams and buses. If you are connected to edoram you can us Internet at university and FH (Haunstetterstraße). Some user are providing free Internet via Freifunk. http://augsburg.freifunk.net/info.html