I agree with JF. Also:
Read the primary schools brochure done by your local authority. Copies will be available from schools and usually online. It will list all the schools in the borough and give information on how many people applied over the ,set few years and the maximum distance away from the school that they took someone.
Go and have a look round your nearest few schools to get a feel for what they are like.
Ask questions, particularly about wraparound care provision if you are going to need it due to work, and local holiday playschemes.
Ask if there are any bulge classes planned. This will again be a boom year and in densely populated areas, bulge classes are put on to accommodate the surplus. These are usually planned a few years in advance (I found out that my local, tiny, difficult to get into church school was down to take one the year I was applying for dd1 and it vastly increased our chance of getting a place), though schools may not be forthcoming about the details.
Read ofsted reports, but take them with a pinch of salt.
If you need to do a supplementary form, make sure you do!
Make sure you are within last year's catchment for at least one school you apply for. Failing to do so gives the local authority the opportunity to place you anywhere in the borough.
As the above implies, in my area it is a real scrum for places.