Oslo is a bit like other european cities when it comes to coffee/tea -houses. I don't know about internetcafés much, but there are some.
Latté-outlets are to be found everywhere, but if you're into that sort of urban lifestyle, get down to grünerløkka or grønnland. That's where most hip and trendy people hangaround and have their ciabattas. There are loads of nice cafés and bars. I'd like to recommend "Fru Hagen". It's rated as the one. But go ahead, take your pick, there's plaenty of choice.
If you're based on the otherside of town, which you're likely to be if you're a tourist, you'll find nice cafés on majorstua and homannsbyen. These areas is slightly cleaner than "the east side", but it lacks a bit of the honest atmosphere. They might also be slightly more expensive. I'd still recommend it though.