Public transport is quite good. The tram lines go into all the neighbouring towns. Within the city, they go 24/7. Fares are about 2 EUR per trip. Tickets are checked quite often, even at night.
There is a camping site in Durlach, which is somewhat expensive though. Just take a tram to Durlach, there is a lot of signs for it. Karlsruhe also has a quite active Couchsurfing community.
There is only one hostel in Karlsruhe and it's rather expensive. Update September 2017: A&O Hostel is located directly at the train station and starts from about 12€.
In the town center, between Europaplatz and Marktplatz there are a some info terminals called blue spots where you can see town maps, send emails and surf the internet for free.