One way to build a sense of independence is encouraging children by recognizing their strengths and allowing them to take on more responsibility around the home that is related to their strengths. My daughter is 9 and cooks entire dinners.
Another way is to give a child age-appropriate choices, where reasonable. Be prepared to live with the child's choice (only give choices between things that are acceptable to you). Start with snacks, outfits, or playtime activities for toddlers, and work your way up from there. My daughter has been choosing her own outfits and dressing herself since she was 2 years old. She has also been insisting on putting bandaids on her own and friends' booboos.
Have patience. Children are as imperfect as adults are. Give them the same consideration and forgiveness for imperfection you would want for yourself and your own foibles. If a child sees that you are honest with yourself about your strengths and weaknesses, he or she will respect you for that and work even harder to please/pitch in.
Hope this helps.