Toddler learning

https://parenting.stackexchange.com/questions/5237/what-should-i-be-teaching-my-two-year-old

These are the things that the Montessori school our son attends looks for; note that these are not things you'd expect a two-year old to already be fully competent in, more that these are a good sample of the items that they measure in their report card:

1. knowing directions (up, down, besides, in front of, behind, etc)
2. body parts (arm, elbow, wrist, eyes, ears, nose, etc)
3. counting (count to ten, count to ten with objects, determine number of objects in a pile, etc)
4. Shapes (square, triangle, etc)
5. Colors (red, blue, etc)
6. following instructions (coloring inside the lines, coloring only the triangles)
7. Singing (they sing a lot of songs)

Other things that he's learned that they don't explicitly focus on:

1. Climbing and physical activity. They have a large playground for him and others to roam on.
2. Sharing. Kids fight over things, and the teachers help them resolve those fights.
3. Injuries. When the kids fall down, the teachers 90% of the time say "You're fine, get up." rather than the motherly 'Gasp! My baby, you've hurt yourself!" This really cuts down on the drama.
4. Finishing tasks. They have and keep a schedule. I have two sisters-in-law who teach kindergarten and early elementary school, and they say that this is one of the big distinguishers between kids who go to (Montessori) preschool and those who don't. Kids who don't get told to finish up never do, kids who do unsupervised learning often have a hard time integrating into a supervised learning environment.
5. Having friends. He knows lots of kids that he wouldn't otherwise know, and he interacts with them all the time. That helps with a number of things, like peer pressure to go to the bathroom ("I've got to go! Rider never uses a diaper!") to giving us weekend plans ("I want to go to her birthday party!").

It sounds like you're focusing a lot on the first set of explicit teaching directives (ie, knowing the basics of language, colors, etc), but not necessarily giving her access to other kids to learn how to socialize with equals and how to function in the classroom. Is that important? I don't know; I do suspect that going to preschool for a few hours a week will help ease her into kindergarten and beyond, but it may be that your daughter is particularly adaptable to new situations and the change won't be so dramatic for her.

I was a teacher in a two's classroom for a couple of years and I have to say, most of what we taught, we taught through play and exposure in books and art activities. We didn't explicityly "teach" as you would see done in a classroom for older children, nor would I suggest such "teaching." Your child is two and will learn simply by being an playing so don't stress out about anything on any of the lists offered here.

Focus on what you see as priorities to you and the day to day needs of your family and family schedule. For many families the two biggest priorities with kids this age are, Safety issues and hygeine, as well as communication.

I was a teacher in a two's classroom for a couple of years and I have to say, most of what we taught, we taught through play and exposure in books and art activities. We didn't explicityly "teach" as you would see done in a classroom for older children, nor would I suggest such "teaching." Your child is two and will learn simply by being an playing so don't stress out about anything on any of the lists offered here.

Focus on what you see as priorities to you and the day to day needs of your family and family schedule. For many families the two biggest priorities with kids this age are, Safety issues and hygeine, as well as communication.

Communication means teaching them how to express there emotions, needs and wants in appropriate ways vs. inappropriate ways. (Kids at this age have a tendency to grab, throw fits, pout, scream or use physical means to get what they want - part of the learning process is modeling for them while correcting) I know you want the cookie, but dinner is almost ready having a fit will not change that - how else you could you tell me you are frustrated/disappointed? . . . you know the typical drill. There is a lot of learning going on in these exchanges alone.

Mostly it is great if you are engaging with your child, facilitating opportunities for your child to play "with" others (which, at this age usually really means play near or around other children) and reading to your child regularly. At two, "teaching" a child is really just about exposing them to as much of the "safe" parts of the world as you can - they will learn from there.

Some ideas I didn't see listed in other answers already are:

Walking in a line (not line of kids, but in a straight line, curved line and zig-zag line).

Name Recognition - child knows and can identify own name.

Matching and Sorting - (you can do a lot at home with this one just while you do chores. Clean up time is all about sorting and what about having her help you sort the laundry?)

Place - children start hearing references to their city vs. other towns or cities he/she might visit. Address is usually learned around four but start using the language with her.

Washing Hands - We even did a whole month all about germs and the importance of cleanliness and hygeine (you are probably doing a lot of this anyway).

Potty Training - Most of my time was engaged in potty training when I was a two's teacher.

Meeting People - introducing, saying hi, nice to meet you. . .

Scissor Skills - this will be important when she does go to school but is often overlooked by parents (I certainly wouldn't have thought of it on my own either). What I mean here, is not using scissors well, just the safety of them, that they should only be used (for now) when supervision is occuring, and how to carry them safely. Kids should get the opportunity to cut with safety scissors starting when they are nearing the age of three - with supervision.

Musical Awareness besides singing songs we did a lot of music and movement, clapping games etc. The kids were exposed to child friendly music in many genres: broadway, jazz, classical, rock, country.

Outdoor Awareness: This included safety issues like crossing streets while holding hands and staying right with guardians and introductions to the basics behind staying away from "tricky people." Outdoor awareness also included naming local insects (learning which ones to alert a teacher to and not touch vs. which ones are just "cool" is a GREAT thing for kids. Just teaching them all insects are something to be careful about is simpler, but creates fears), trees and other plants (not that we had any in the classroom, but learning that some plants are safe and others are not for touching is similar to the lesson about insects), commonly encountered large animals (and safety around them) as well as seasonal awareness (changing leaves, weather etc).

Community Helpers: becoming familiar with our uniformed public and their respective jobs (police officers, fire fighters, paramedics etc.) If there was ever an emergency in which your child needed help from one of these people it is good for them to recognize the uniforms and job of that person.

Sensory Activities: Sensory Stimulus can be a natural part of growing, but it can also be limiting to kids if they don't get enough variety because they can develop fears of the unknown - it also helps with writing skills, oddly enough, and some such activities can be major stress reducers - working with playdoug is an example of this, but my favorite was to squirt some shaving cream on a smoothsurface and just let them play in it, play with feathers, leaves, beans, water etc all count as "sensory" experiences.

If you would like a short list of what to expect in terms of developmental stages in your child's learning PBS has this to say about it.