Well, this is a very simple thing to say how to do and not that easy to get done. But there are a few things that you can do and which I have done. The very first thing, is you are a new blogger, after you create your own blog and have atleast a few blog posts published, is to look for blogs that you would want to read more than just a handful of posts. One way is to use google for the topics or search for blogs in the particular niche that you are interested in. If it is not a niche but bloggers from a particular country, state or city that you are interested in then do a search for that. There is always a list for them.
Once that is done you start reading their posts and if you find that you enjoy reading them, start commenting on them. Ofcourse add your name and blog URL in the comments so that they can easily check your work out. Then create a blogroll and add the blogs that you have liked and keep adding to them until you have a good number. Keep reading their posts and commenting and replying to their comments. Ask them to check out your blog and when they comment reply back to them. Soon you will have atleast 10-15 (after you keep visiting maybe 30 to 40 blogs) and now you have a small group of bloggers you have a relationship with.
Point to note, a lot of the bloggers I have create a bond and friendship with have stopped blogging altogether and instead use Facebook & Facebook groups to get their posts across to their friends. Some also use Twitter. I am friends with most of them still so I am happy that we keep in touch. A few seem to have disappeared. It happens.