The fat in your breasts is normal, they will feel lumpy and sort of hard. What you need to watch for is new lumps that were not there before type thing. If you keep up self exams and monitor how your body is then you should have a good indication when something is not right.
Those things you are squishing...are pimples. Some big, some small, there is a complete variety. The white stuff is the infection, the pus. No worries though, they happen to everyone. Just keep your skin clean and free of extra oil build up and they should clear, try those oxy pads that are for the face. They will clear up your pores and help keep the tiny infections away.