My suggestion to you would be to backup all of your blogs you follow, visit each blog and simply bookmark it so you don't loose them. Then try a few blog readers before July 1st and decide which one you like the best, and then you will be all set and not left scrambling on July 1st.
Below is a screenshot of Google Reader. I have removed some information for privacy but basically this is what it looks like. You can organize your blogs into folders on the left. I have everyday blogs which are blogs I read first before reading the others. As I have a hard time keeping up with everything, so it works better for me this way.
In the main part the blog posts appear. Once I scroll past a post, it marks it as read. If I want to comment I click on the title and it takes me to the actual blog.
~ click the images to see the full sizes ~
Some have been wondering if they use Google Reader, so here is a screen shot of the Blogger Dashboard's Blog Reading List. It is similar but a little different.
I have just started using Feedly, which is a blog reader as well. So far I really like it. You can customize the theme, and you can tweak the settings to fit your preference You can see your posts in a titles only format like Google Reader lets you, or you can choose to view it magazine style, which is what I have it set to... I kind of like it that way so far. It also lets you follow your youtube channel subscriptions, although I haven't tried that just yet.
If you would like to try Feedly, you can find it at http://www.feedly.com.
And you can do a search for Blog Readers to find others!