Do you have comic shops in Sweden? I think you're in Sweden. If so, go. Talk to your comic shop guys. Tell them what kinds of things you enjoy out of stories, settings, plots, etc. Tell them what you don't enjoy. Tell them what titles tickle your fancy and why and what ones you think you want to avoid and why. Tell them your budget and what you hope to get out of your reading experience. See what they suggest.

If you're looking at reading cape comics and you know that you don't want to invest a lot of time and money into following anything in particular, ask them about stand alone stories that they'd recommend. For example, if you're interested in Batman, they might recommend the graphic novel The Long Halloween. It's a great Batman story, disconnected from the Batman universe as far as continuity is concerned, and has a large cast of characters that you can enjoy and appreciate without even knowing The Batman comics too well.

If you're looking at comics as a medium and aren't too interested in following in particular, ask them about standalone graphic novels they might recommend. There's stuff there all over the genre spectrum from science fiction to drama to mystery to crime to autobiographies. There's stuff out there that spans from the mundane to the surreal to the avant garde.

If you're really new, and really interested in comics as a medium, before anything else, I recommend you check out Understanding Comics by Scott McLeod. As crazy as it sounds, reading comics well requires a little understanding of how the medium works, and that book will help you be able to dive into comics headfirst and allow you to appreciate them than if you just went in all willy nilly.