I've recently found out I have to do US tax returns as a dual citizen of the USA and the UK. I've also recently sold my property in the UK, and am about to move into a new property in the UK. Do I have to pay tax on all my gains from the property, or is there tax relief as this is my main residence?

Even though you may need to do US tax returns as a dual resident of both the US and the UK, nevertheless you need to determine, using the double tax treaty ‘tie breaker’ rules, where you are tax resident - see Article 4 in www.gov.uk/hmrc-internal-manuals/double-taxation-relief/dt19903 and 19933. If you are tax resident in the UK, you only look at the UK rules for taxing the sale of this UK property. If you are deemed to be tax resident in the US, then since this property is located in the UK you must first pay any tax in the UK according to the UK tax rules, and then pay any tax in the US while claiming double tax relief for the amount already paid in the UK (Article 13).