The two answers already posted give some great advice about choosing a district to land in, but there are specifics about each school you can also choose to take a look at - part of that, "Thoroughly examine the final candidates" step mentioned by yair (Great synopsis of the process btw) Some of the items I've listed here, might be a bit nit-picky, but they are the kinds of things a lot of people don't know they can even inquire about and can make a huge difference for you just in knowing about them (you can decide if something is a deal-breaker or not, or if you don't like it, but are willing to live with it, you can figure out your plan for "dealing.") Most of these items are helpful to know whether you Go Private or Public but a few may be more applicable to one or the other.
Know and understand what the school's mission statement says. Part of accreditation of private schools in the US includes having a mission statement, many public schools also have one. This expresses the general beliefs about what the goal of education is within that school community. Not all schools pay much attention to their mission statement, but you should have a general idea of what the school publicizes as its goal for your child.
Know homework standards. Some of the specifics will be dependent on specifically which class your child is in. However, Principles - especially at the upper grades but increasingly even in Kindergarten - often give minimum amounts of homework that should be assigned to kids - even when the teacher feels the class has fully understood a lesson they are mandated to assign a certain amount of "practice work". Know how much you can expect for an average week and an average night
Know the school schedule and attendance policy. This may seem as though it is an obvious one, but which holidays the school decides to offer time for, how often there will be half days and when there will be additional Fridays or Mondays off can be revealing about what a school expects from parents and community in terms of time and commitment. For example, if you take a family vacation during school sessions can your child expect to get make-up work in advance? Is there a difference between kids leaving because of bereavement vs. family holiday? etc. For each day missed, how many days are allowed time to make-up the work for full credit? Is there a difference between an absence for illness and an absence for other reasons? What about Tardiness?
Know and understand discipline policies. Even if your child never misbehaves they can be affected by these policies. For example, how likely is it another child will be able to push, hit or otherwise demean your child and get away with it? At one school where I worked "redirection" was the standard policy for discipline. This means, when a child misbehaved in some way he or she was to just be distracted with a new activity to do. While this works most of the time (in Preschools it is an especially common standard) the result when it was strictly adhered to was that children were simply redirected even when they were physically aggressive toward other children repeatedly.
A lot of stock is put in ratios. Ratio of adults to children DOES matter a lot. The optimum ratio (for all grades really) is about 15-30 kids (although at the elementary level anything higher than 25 is too much). If you go much lower than 15 school-aged children, the energy in the rooms is draining, much higher and you can't truly get to know each individual and their learning styles, strengths and weakness. However, ratios are often misrepresentative of what is actually true in a given school. It is an average schools produce for the information of the families looking into them. Ratios in the standard classroom can be higher than what is published because enrichment classrooms which may have a ratio of one to eight are also averaged in. Other times, a ratio might be given as an over-all. These ratios are purely the number of staff (including support staff that is not in the classroom such as the receptionists), faculty and administrators. Sometimes staff is left out of this ratio, but administration almost never is. Almost every school I have interned or worked at has had ratios that paint a picture that is very different from the actuality. Since this is true overall, when comparing a ratio at one school to another school it is still valuable information for you, just make sure you know that in most schools, a ratio of 20 to one does not actually mean your child will be in a class with no more than 20 kids per adult within the classroom.
Know policies for special needs - you may have a child who is academically gifted and will need a more challenging or specialized curriculum - you just don't know it yet, or you may have a child that is really smart but his/her as-yet undiagnosed dyslexia or dysgraphia makes it very difficult to read so your child will need extra reading supports. What is the process for determining a child may need extra supports and how long does it usually take? Are there in-classroom adjustments that are made for fast or slow readers? What is the school's policy on holding back or skipping children? I have a niece that is particularly advanced in a couple of subjects, but rather than skip her she is given work about the same topics the other kids are studying that is deeper - or more challenging. She is NOT given EXTRA work but work that is appropriately challenging for her. What a healthy and advantageous policy for her in her educational journey and one we couldn't get in our region for our daughter.
Know about additional fees and incidentals. If your child attends a private school there are often book fees, uniforms, activities fees (for field trips etc.) that are not considered part of the tuition. Even at public schools there are additional supplies the school does not supply, field trip costs and fees for extracurricular activities. Get a look at the required supplies list no matter whether you are at a public or private school. I've heard complaints from families that spent hundreds of dollars on supplies because they needed particular type of notebook, calculator etc. etc. While the specifics of a list are likely to change before you are at the point of shopping for these supplies, the size and scope as well as required specificity can be an indicator to you of what to expect.
You might also look at sample report cards to know what will be considered important in terms of grading at different grade levels and to know what the gist of formal evaluation is aimed at in your school community. Some schools put a lot of emphasis on effort while others focus more on scores on homework and tests. Many really don't emphasize any of the subjects except math and reading until much later grades while others give scores for all subjects studied no matter student age and level.
Know ratings scores. The public and private schools (in the US) all administer standardized tests for the purpose of evaluating the success of the school itself. Too much stock is often put into these scores in terms of evaluating teachers and administrators as these tests usually only focus on reading and math at the elementary level (science is added in 4th grade at public schools in our state) and they don't consider factors that the school has no control over such as how many kids are in the school that don't have a computer at home, whose parents don't read to and with their children, or how many of their students come to school hungry every morning. However, ratings scores for schools factor in test scores as well as other elements like extra-curricular options that are available, drop out rates (in upper grades), building maintenance and care, etc. When you compare the school's current score to its scores from previous years you can see whether the school is improving and growing or whether it is faltering. It also gives you an idea of how one school preforms compared to another when you are able to choose between two ore more school options. At this time, schools are expected to show a rise in performance each year or their scores do go down slightly so a consistent score does not demonstrate stagnation. In the public schools, a score of 800 is considered acceptable performance by the state in which I live, you may need to do a little more research for your state.
Know policies about what is or is not allowed at school. I've known of more than one story of a child getting in trouble for the butter knife that was packed in a lunch pail by the mother. Many schools (especially in the upper grades) also have policies about cell phones and other electronic gadgets
Don't forget to ask about environmental and safety standards as well. Pesticides and Herbicides are often used extensively on school grounds and athletic fields. What are the safety standards? How long does it usually take for a needed repair to take place? What hazard exist for your region to consider? What are the fire procedures? tornado procedures? Earthquake procedures?. . . Even if the answers aren't make or break for you, asking these questions lets the school know you care and are conscientious. It may even lead to improvements in school standards if enough parents start asking these questions.
Don’t forget to check out the cafeteria and make sure you are satisfied with the "hot lunch" offerings and their nutritional value. Look up Jamie Oliver for more information about evaluating school foods.
As an added comment, you don't bring it up in your question, but I'll share that we had a really hard time finding a school for our kid as none of the schools in our area are all that great. Even the private schools didn't seem to match our needs or intentions so we wound up going with "virtual schooling" which offers a combination of public schooling with classrooms and the whole nine yards once/week, online classes she does at home but who have a teacher other than myself and homeschooling. It is just another option you might consider if you do truly have a hard time finding and getting your child into the "right" school for you. I have asked a question here about what to think about when considering whether homeschooling (or any form of it) is right for you that if you are open to the idea may also be enlightening in regard to both the downsides of choosing this route and the advantages.