What tasks (and at what frequency) should a stay at home mom be taking care of?

That is hard to answer without knowing what state of affairs you like to keep your home. Some chores are easier if you do them daily, like vacuuming and laundry. Some chores are good for two or three times a day like dishes and meal preparation. Usually dusting can be done once a week. Cleaning the bathroom depends on how quickly it gets dirty in your home. I raised two boys so sometimes I would tidy the bathroom everyday and give it a thorough cleaning once a week. Moping floors can vary too, any where from daily to once a week.
The children can clean up their toys every time they are done playing with them, probably several times a day. Change the bed sheet once a week unless they are soiled more frequently.
All in all if you spent 2 hours a day on chores and don’t let them pile up then it is much easier.

Routine helps. I’m a SAHM to an almost 3 year old and one on the way. If you stay at home by choice it’s your job to care for the home which means the house cleaning, laundry, children and meals. I’ve always found breaking up the big jobs through the week (laundry for example) helps. I vacuum twice a week (one at the beginning and one towards the end of the week). Things like dishes, feeding pets, child care, making beds and things that have to be done every day get done through the day. If your child sleeps during the day take advantage of that time to work on household things.
As far as organization goes we have a family calendar with appointments, social engagements and comittments on it in a prominent place in our kitchen so both my husband and myself know what is going on week to week at a glance. In the playroom I have a bin and shelf system for toys to make clean up easier and the mess at bay. Grocery runs are done biweekly for my family and I put meals that use things with sooner expiration dates ahead of those that don’t OR freeze it if possible if I’m not going to use it right away. My husband works full time at his job but even when he’s home he helps out with doing dishes occassionally and he takes care of the outside stuff like the lawn, trash each week, cleaning the garage out when needed and heavier projects around the house that I know I am unable to handle. He takes his time as well staying with our son some afternoons so I can run errands child free when needed occassionally as well as helping out with the routine of caring for our child. Pet care falls on him for our dogs and I handle the feeding and care of our two indoor kitties. He’s even known to take over cooking on weekends or other times to give me a break in the kitchen now and then through the year and in summer when he’s not travelling for business he usually takes over the cooking exclusively for me in the evenings so we don’t have to stand over a hot stove or oven inside. When I have time I may bake something as a treat or work on projects for me that I enjoy doing, but the priority is my child (soon to be children), the household upkeep and then myself.