I'll answer your question, even if I think it is a bit too much general.

Can you give me some tips about how to read the papers, until I find my own way of doing it?

It depends on what you are interested in. I would start by reading some reviews and perspectives on the topics you like more, then look at some article to see how people carry out the work.

Can you give me some places to start my readings?

You can find a nice list of journals related to bioinformatics here. See also some questions posted here on Biostar: which journals do people read, which blogs.

how do you organize the papers you are reading?

Mendeley is a very nice software to organize collections of articles. Since I have been using it, I have saved a lot of paper because it makes it comfortable to read articles on the screen, as it allows to put notes, highlights, and synchronize your collection on multiple machines.

My first question would be: do you have a genuine interest in pursuing bioinformatics, or is it a vague notion? I'm not questioning your motives, but I wonder how you arrived at the idea of studying bioinformatics, given that you know little about it and were not exposed to it during your undergraduate education?

I don't think that you will learn effectively by trying to read academic papers, without some basic education. Learning how to read papers is a skill to which you should first be introduced as part of your formal education. So by all means take in interest in bioinformatics but start at an appropriate level: find some basic introductory material on the web, seek out advice as to what courses you should take and where to take them. Leave the papers for that Masters course.

My final tip: there's a "bio" in "bioinformatics" for a good reason. A good basic knowledge of biology is at least as important - perhaps more so - than computational skills. Again, this should be a component of whatever graduate program you decide to pursue.

It really depends on what you are interested in .... bioinformatics is a very wide subject. I'm not sure about the "how to read a paper?" question ... bioinformatics papers can be very heterogeneous. But I'd start off by having a look at some of the articles in PLOS Computational Biology , it also has a "Getting Started series" which might help.

If your university doesn't do any courses on bioinformatics then maybe looking at MIT Open Courseware might be useful for you 6.092 Bioinformatics and Proteomics

Organising papers .... I am a mac boy so I use Papers ... its quite nice... although I've heard people say that Mendeley is good. In you want something more internet based I'd recommend Connotea

Bioinformatics is now a huge field. what are you interested in? Trying to learn bioinformatics by reading papers is like trying to learn to program by reading programming books--it's just not efficient, you have to have a project that you want to work on. So, maybe a good way to approach it is to just read review articles or seminal works until you find something you're interested in. Then, start a project and that'll force you to read related work.

I've had several people ask me somewhat similar questions. You can see the questions and answers in this blog post:

http://cdwscience.blogspot.com/2011/08/what-is-it-like-to-be-bioinformatics.html

I think there are a lot of factors beyond reading papers. I would probably say the most important other factor is looking for what jobs are available for someone with a bioinformatics degree.