You'll have to find out from your school district office what the options are in your area. Some districts do offer interdistrict transfers. In my experience it's really hard to get one of these granted because of space limitations at the more desirable schools.
If your school district has any magnet schools that may be an option for you. Magnet schools are still public schools (no cost to the parent) but they don't pull kids from a particular neighborhood. Anyone who lives in the district can apply and usually there's a lottery to determine who actually gets to go. It may be too late to apply to magnet schools now for next fall - but maybe not!
Of course private school is always an option but that's obviously tough financially.
A final option may be trying to transfer to a school in a different district. California allows this but I don't know about other states. You may run into the same problem of having the best schools being too full to accept transfers.
There are several things you can do. In my opinion, a good kindergarten should a half-day program (not full day), have centers, have no more than 10 to 12 children, and caring instructors. The facility should be very clean and positive ;)
There are many kindergartens around both private and public. You can call and observe the ones you might be interested in.
I personally recommend a Montessori kindergarten. This is a hands on learning environment that teaches children a lot of the basics that are sometimes missed (tying shoes). They cover colors in depth, shapes, tactile learning, and a lot more. I do not recommend a kindergarten that uses technology as a means of primary teaching. Children need "hands on " at this stage. Good Luck :)