The left side of one of my breasts is sore to the touch. Should I be concerned?

Probably not. Since you are young (27), it is most likely you have a cyst, fibrocystic breast disease (painful lumpy breasts), a bruise or an infection. If the pain does not get better after a few days, or if you develop redness or a lump in your breast or under your arm, see a doctor. This is very unlikely to be cancer!
Yes or no. Most things that cause pain aren't bad and most of the bad things (ie cancer) don't cause pain until more advanced and obvious. Some degree of breast discomfort is very common and often multifactorial -benign (fibrocystic) breast pain and/or chest wall/rib related. If there's an assoc lump it could be a symptomatic cyst that can be relieved w/aspration. If you're concerned see your doctor.
Perhaps. Any new pain in the breast warrants some concern. It is reasonable to wait one menstrual cycle to see if the pain goes away; if not, i advise you to see your primary care doctor or gyne. Of course, if your concern leans towards "worry", it may be reassuring to you to go in sooner.