I have dual citizenship in Canada and the UK. How long would I be allowed to stay in the United States under both visas? Am I able to use the two visas concurrently and, if so, what procedures are required to leave the USA and re-enter after a few days?
British citizens can enter the US for up to 90 days at a time on the Visa Waiver Program (VWP). If entering by air, they need ESTA authorization.
Canadian citizens can enter the US for up to six months at a time without needing a visa. In addition, the do not need ESTA authorization, because Canadians' visa-free entry is not related to the VWP.
You cannot stay in the US for 9 months straight by coming for six months as a Canadian and then going to Canada for a short while and reentering for three months as a British citizen. If you try to do that, and are found out, the immigration officers will be unhappy, and you will most likely be denied entry.
Your best bet is to enter as a Canadian, and just work within those restrictions. If you spend more than half a year in the US, though, be wary of becoming a US tax resident; many Canadians apparently have to pay close attention to this when they travel to vacation homes in the US.