Welcome to St Louis!
As you can see there are lots of people available to help you, but most aren't addressing your actual question about neighborhoods. St Louis ranks as one of the largest metropolitan areas in the US. That said, we still pride ourselves on our "neighborliness". Like all large areas, we have some areas that have a higher crime rate than others and we have homes in all price ranges. We are considered one of the most affordable cities in the US, when they compare the average price of our homes to the average person's income in our area.
To a Realtor, your question is a bit vague though - everyone judges safety, affordability, school district ratings, walkability, etc through the lenses of their own past experience. What one person might consider affordable might still be high for another person, or it might be a drop in the bucket for some of our wealthier homeowners.
A good starting point when you are looking at homes in a new area, is to focus on something you want to be near - for some it's work, others may choose their place of worship, school districts are often looked at, family or friends in the area, whatever is important to you. I had one young man who wanted to be close to the gym he regularly went to.
At the same time that you are considering areas, you should be talking to a mortgage lender to find out what you can qualify for. It is often best to find a local person who can help you, since they will know if there are any special grants or loans available in the area and they will be able to come to closing with you. If you dont know a local lender, any of the agents who have responded here, will be able to refer you to one. You should ask the lender to run the numbers 2 ways for you - what is the price range that they will be willing to give you a loan at, then tell them where you are comfortable with having your payments at and ask them to run the numbers backwards and let you know what that value is.
Next let whichever Realtor you decide to work with put you in a special online search that is tied directly to our MLS, where you can change the search criteria, put notes with houses, check maps, etc, based on what you have decided from doing steps 1 & 2.
Along with that, make sure to sit down with the realtor when you get to St Louis to discuss how our housing market differs from the area you are coming from
If you have any questions or would like to discuss any of the points I've made, please contact me directly through my Trulia profile or at DWeir@CBGundaker.com.