Checking the upper outer part of the chest towards the armpit is especially important, since about 2/3 of breast cancers develop in this area and there's also a lot of lymph nodes in this area to check out.
Look for changes in the skin, like nipple inversion, redness/duskyness, tenderness, or a dimpled appearance like orange peel.
I hope this point is obvious but don't do a self exam until everything is FULLY HEALED after surgery. When they say firm pressure they mean enough that its uncomfortable on healed tissue. If you press that hard on an area that's healing you're going to regret it.
If you have had top surgery, you can also talk to your doctor about screening MRIs, since mammograms aren't exactly feasible, but insurance coverage for that might depend on your family history, age, and other factors.
If you're under 30, unless you have a high rate of breast cancer in your immediate family, self-exams won't do any harm but they also aren't really needed. Most people should start in their mid 30s.