You are probably weak. Strength training is the solution.
The best option would be to learn to lift weights. Starting Strength, a 3-times-a-week barbell program, is a good option. StrongLifts 5x5 is also commonly recommended. (See this questionthis question.)
If you can't get access to a gym or barbell, you could look into bodyweight strength training instead. These twotwo questionsquestions have some good resources on that front.
With More Information, You Get More Advice
Since you gave us your diet and dimension info, I now more firmly suggest strength training, and would add:
- BMI is not a valid metric for individualsBMI is not a valid metric for individuals. Please don't misuse it, or allow others to misuse it, by applying it to people instead of populations.
- Why do you want core exercises? I hope you have not fallen prey to the distressingly common virus called Believing in Spot ReductionBelieving in Spot Reduction! You can't get rid of belly fat with crunches or any other targeted exercise.
- I am not convinced that you have a weak core, since you haven't stated any reasons for believing so. You might be weak generally, but as bodybuilders say, everywhere is a weak point if you don't lift.