Core Excersises and Weight Loss

Ok so I'm a 17 year old multi sport athlete. And really I have two questions here. One, I'm 5'10" and weigh 178 and am very insecure about my weight due to comments by my family members. Do I need to lose weight? Or am I at a healthy weight? And then also does anyone know any good core exercises? I have a very weak core and it's starting to give me back problems so I'm trying to strengthen it. Any comments about wither question is greatly appreciated!