Silvertap,
I hope that my post doesn't come across as 'teaching my grandmother to suck eggs', but I know from my DH (no-one in his family went to state schools) how it can be hard to get one's mind round how a different sector 'works'.
Firstly, on 'choice of schools'. For a state primary, although every area of the country allows you to apply to multiple schools, there is frequently not a genuine choice. For many people, the choice comes down to their nearest or catchment school, or an unpopular or isolated one a long way away. For a few people in urban centres, there may be no local school that you are guaranteed a place in.
So whereas for private primaries / preps, you can visit them all and choose one 'as a customer' (though of course there are some which have competitive entrance tests), for state primaries, the first step is to identify which schools you might have a chance of getting your children into as an applicant. The criteria will GENERALLY include distance, but if you are in an area with many faith schools, church attendance / affiliation may also be relevant. If you look on your council's website, they should have an online copy of the admissions guide for the current or last year, which will include all oversubscription criteria - ie how they will allocate pl;aces if there are more applicants than places.
www.schoolguide.co.uk/, if you enter your postcode, is a useful (though not 100% reliable) guide to the schools close to you, and will give an indication - which should be taken with a pinch of salt - as to whether someone living in your address might be within the admitted distance from that school. It is obviously less reliable when it comes to schools with e.g. faith-based admissions criteria.
Secondly, once you have narrowed down to the schools that you have a chance of getting into, you should a) read their Ofsted report and b) ring them and ask to visit on a normal working day.
On Ofsted reports - look VERY carefully at the date. The criteria for the different gradings have changed quite dramatically over the last 5 years, but many schools graded 'Oustanding' more than 5 years ago have not been reinspected. the criteria for an old 'Outstanding' would barely get you a 'Good' now, and many of those schools have become quite complacent in their outlook as they are unlikely to be inspected. A recent Outstanding is more likely to be genuine BUT all inspections are highly data driven, and so a 'Good', or even 'Requires Improvement' school may well be a more interesting and 'striving for better' all round place to learn than an 'Outstanding' one.
On school visits - I have never worked in or visited a Primary where the head has not personally escorted prospective parents round the school on request on a normal working day. Some very popular primaries do try to group parents into workable groups rather than doing individual tours, just to make the process slightly less time consuming. However if tours are not available at all and you are pointed towards an open day, I would worry. You want a school where the head is open, welcoming and wants your involvement, not one where they seek to keep you 'on the outside'.
On 'what you might see' - and what you might not. In state primaries, classes are normally taught in a single classroom by the same teacher all day, and there are not subject specialist teachers nor specific rooms for e.g. science. If this is important to you, then that is more often found in private schools (although even there it is worth asking whether the person teaching the class in the 'science lab' is actually a science teacher, or just the normal class teacher, who might have a totally different specialism). All schools will teach indoor and outdoor PE, but the assembly hall will usually double as the gym, and a local park may serve for outdoor football pitches.
In the Reception class, state and private teaching practice may well differ. Some private schools are very 'formal' from the very start, with lots of 'sitting in rows and learning' [there was a very sad thread a little while ago about a summer-born boy who had had day-long exclusions from his private Reception class because he couldn't sit still through double Geography]. In pretty much all state schools, the Early Years philosophy of 'learning through play' is the norm, so your initial impression of a Reception class might seem chaotic. Do spend some time watching - there is a genuine difference between 'groups of children marauding around aimlessly' [avoid] and 'busy, purposeful play, gently guided by adults'.
On secondary schools - it is worth looking at which school(s) you might b in catchment for, but not worth visiting them at this stage. You could check their admissions arrangements to see if they have any named 'feeder' schools - rare but not totally unknown - as part of your primary school considerations. It is also worth asking the primaries you visit where their leavers typically go on to.
On the 'all round package' between state and private. One of the advantages of the private route is that it is often possible to 'buy the package in one go' - to choose a school where your child can pursue both their academic and extracurricular life. To build that 'all round package', a state school parent may well have to find other external providers of some of the extracurricular areas. The advantage is that you have a wider choice - the v. serious dance class vs the easygoing one, the youth wing of the professional football club vs a Saturday kick around with friends, the music teacher who loves jazz vs the one who focuses on strict classical technique - rather than the 'one size fits all' version that comes with the all-in package. However, it requires more parental input, and if you already know that you will need 8 am - 6 pm care and would love much of your weekend to be free, then it may be easier to buy the 'all in package' rather than the logistical pressures of getting your child to dance / music / county orchestra / football training after school or at the weekends.
Academically, the balance between state and private depends ENTIRELY on the specific schools. If you live in an 11+ area, then private primaries have the advantage that they can coach for the 11+ from the age of 4. State schools are not allowed to do any more than the most basic familiarisation. The vast majority of private schools actively manage out or never admit children with SEN, and so their 'raw' results may look better - but so they should, given the restricted range of ability on entry.