You must take a COVID-19 test on or before day 2 after you arrive. No quarantine needed. Only need to quarantine if NHS Test & Trace informs you that you travelled to England with someone who has tested positive for COVID-19.
No need to quarantine if you have been fully vaccinated. You still need to book a day 2 test to take when you arrive in England. If you are not fully vaccinated under the UK vaccination programme, you will have to quarantine on arrival and take both the day 2 and day 8 tests.