Can pain in my side be a sign of pregnancy?

Yes. Sometimes it's ok to have pain during pregnancy, but sometimes it's not ok to have the pain. For example, some pregnant women have a cyst called a corpus luteum cyst, and that can cause some pain. Others can have a tubal (ectopic) pregnancy, which can be life-threatening. It's imperative to be tested to see if you are pregnant, and, if pregnant, what the pain is from asap!
Maybe but unlikely. In general pain anywhere is not a sign of pregnancy, unless it is one in the wrong place called an ectopic pregnancy. Usually the first signs of pregnancy are a change in one's period, usually stopping bleeding (but not always), swelling or enlarging breasts and nausea. Severe pain should be checked out.