So how do you know when your kids are ready for independence? All kids are different, but here are some general guidelines:
Yes, infants! Allow your babe to do as much as she can. When she is able to reach for and grab a toy, don't just automatically pick it up for her - let her try to reach for it. When she starts to try to finger-feed, let her! When she grabs for the bottle or sippy cup, let her hold it! When you really think about all the things an infant learns to do in the first year of life, it is quite amazing - so let her learn and do!
This is generally the time when kids most want to be independent, particularly if they have older siblings to imitate. So, just go with it! If she wants to walk on her own, let her. Encourage her to fetch her own clean diaper and wipes for changing time. Encourage the exploration and use of utensils while eating, and start introducing a small open cup (I suggest a Dixie cup or even a shot glass to minimize the inevitable mess). She is probably helping with undressing and dressing (by holding out her arms and legs for you), but start showing her how to push down her own pants, and later how to pull them back up. You can place your hands over her hands to show her how. If she's interested in the potty, go with it (but don't force it if she's not ready).
Your child's independent skills develop exponentially in this stage! In the third year of life, she will be ready to feed and dress/undress herself. (Note: In some cultures, the parents continue to do these things for their children through this stage - do what is best for your family.) As for the potty, most kids become potty trained around age 3, but if your child is not ready, don't force it. If she is potty trained, allow her to do as much of the process as possible - pushing down her pants, sitting herself on the potty, getting off the potty, pulling her pants back up, and washing her hands. A small stool like this or this will make things easier! You will probably still have to help with wiping for quite awhile (most parents say even their kindergarteners do not wipe well)! Encourage your 3-year old to start putting away some of her toys after playtime, putting her dirty clothes in the hamper, and putting her shoes in a basket - just don't expect a spotless room!
Your 4- and 5-year-old will add simple household chores to her repertoire, like putting away most of her toys, feeding the pets (with guidance), spraying/wiping tables (with water or kid-safe cleaner), and helping to set the table (unbreakables!). She can also be encouraged to brush her teeth (you may have to help with squeezing the toothpaste if your kids are like mine and make a huge mess; and you should also finish the job for her to ensure clean, sparkly teeth), brush her hair, and wash herself in the shower/bath (with your supervision). Encourage her to get her own drink and snacks.
School-age children (6+)
Just keep adding on in this stage! Household chores, helping to care for younger siblings, making small meals, caring for small cuts, ordering at a restaurant, making small purchases, saving money, etc. If your child is capable of doing things for herself, and of helping you around the house, encourage that!
Encouraging independence takes a bit of effort from you in the beginning, but it pays off (in your time and sanity) in the end!