Short Answer? Get rid of the Tories.
Long Answer?
1. Get rid of the monarchy. I'm actually a soft royalist but if we're going to get rid of the class system we have to start at the top. Set up a charity to look after royal estates for heritage and tourism purposes.
2. Give Scotland an indie-ref and abide by the result. If there is sufficient support also Wales.
3. Split England up into its historic pre-norman Kingdoms (Mercia, Wessex etc) as a series of swiss-style principalities with their own local governance. (I would call this the 'Heptarchy Act' and frame it as a return to the idea of historic Anglo-Saxon England. Fold this in with removing the hereditary descendants of William the Conqueror from power)
4. Devolve regional powers to the principalities with defence, currency etc left to the national government.
5. Introduce proportional representation at all levels of government.
6. Abolish Lords. Replace with 3 upper chambers sat in historic locations: York, London and Winchester. Members of the public would be chosen at random to sit in the upper chamber in a manner akin to jury duty. These would sit when an issue or act effects the North, SE or SW respectively. This is to encourage more local engagement and make sure people feel their voice is being heard.
7. Rebrand England as the 'school of the world' and leverage our world class higher education system on the world stage. Create education partnerships with other countries: in return for reduced prices for their citizens they would offer places to our own young people which for lower income families would be fully state funded. This would allow children in poverty to study abroad and widen their horizons at no cost to themselves.