What a Personal trainer should do.
#2. Address any postural quirks (teach you neutral spine and stabilization) and modify exercises for weaknesses and/or past injuries.
#3. Design a full program for you, not just weights, but cardio and stretching as well
#4. Assess your current level of strength, cardiovascular fitness, flexibility and monitor your progress.
#5. Update your training program every few weeks or months.
#6. Educate you enough so ultimately you don't need them anymore. (A really good trainer will "graduate" you so you can work out without them!)
#7. Be mindful of your individual goals, since each person is different, and goals can vary.