The education system in the UK is different depending on whether you are in England, Scotland, Wales or NI. I'll assume you're in England, as it's the most populated.
Children have to stay in school till 16, after that they are required to either stay in full time education, or start an apprenticeship/traineeship, or do work/volunteer for > 20 hours a week whilst also in part time education/training.
Whether or not schools have post-16 provision (often referred to as a Sixth Form) is very varied, there's no requirement for schools to offer it. If there are no school Sixth Forms in your area, then there will be Further Education colleges, which offer the same kind of post-16 education (A Levels, and similar) but in a slightly more University-like environment.
Schooling in England begins with the Reception class, not Year 1. Primary schools cover the Reception year, then Year 1 through to Year 6. Secondary schools start with Year 7 and go up to Year 11, then if they have a Sixth Form they will include Year 12 and 13.
Some Primary schools are split into separate schools for Infants (Reception to Year 2) and then Junior school (Year 3 to Year 6). In my town, there are mostly Primary schools, with a couple of combination Infant/Junior schools.
Nearly all state-provided schools are not selective, so there are no selection tests at any point up to age 16. In some counties in England, there is still the old-fashioned grammar school system for the state-provided schools. In those counties, pupils sit an exam called the 11+ at the end of primary school. The top section of pupils will be eligible to go to the grammar school in their area, and the remaining children go to secondary schools instead. Parents typically are desperate to get their children into the grammar school and will often hire private tutors and do extensive exam prep with their child.
Whether or not your child can continue into Sixth form (or go to a FE college) depends on their GCSE results, and each school/college will publish their entry requirements and application process.