By now your 6 year old can be given small simple responsibilities like making her bed, hanging up her towel, watering plants, setting the table, and cleaning her room, to name a few.
Assigning age appropriate chores will make your child enhance her self esteem and help her feel part of the family "team".
What you can do
To start off, make a simple chore chart where 2 to 3 tasks are listed; for example, setting the table and making her bed. First, show your child how to set the table step by step. Next, let your child help you do it. Then she does it alone under your supervision. Once she has mastered it, she is ready to do on her own.
Set realistic expectations initially. Through encouragement and appreciation, she will feel good about her accomplishments. Add new and bigger tasks as she gets better with the old ones. For instance, if she has mastered making her bed well, give her the task of keeping her room clean, wherein you explicitly mention keeping toys in the toy box, clothes in closet, books on the shelf etc. Over time, your child will learn the skills that she will need as an adult, and at the same time learn to work together as a family.