You definately have fungus on your plate. Althought it's hard to tell what exact type of fungus it is I'm not an expert. I'm assuming those are blood agar plates? Follow the steps below in order to get better results for bacterial growth. It sounds as if your plate preparation needs work.
Once you make the agar you will want to autoclave it (or sterilize it somehow, pressure cooking is a good option if you don't have an autoclave). Once the agar is autoclaved you will want to wait until it is cool enough to touch, it is better to let it cool in a water bath (usually mine are around 55 degrees C). Then if you can light a continuous flame (like a bunsen burner) by the agar and the sterile petri dishes. Then just pour the plates by the flame. Then let them cool until they are hard (or at least not fluid-like) but don't put them in the freezer (I'm not sure why, but my professors have always told me not too). Just remember that nothing is sterile around you except the sterile petri dishes and the autoclaved agar (not even the air which is probably where your fungus came from).
Sorry this is so long.