A visit visa to Dubai can be obtained by British passport holders on arrival for 30 days at the airport from the immigration authorities. Additionally, you must get a tourist visa if you wish to stay in the UAE for longer than 30 days or if you have used your 30 days within the last 180 days.
As a UK tourist, you can visit Spain without a visa. However, you must abide by the 90/180-day rule, which means that you can stay for a maximum of 90 days within 180 days. It's important to keep track of your days accurately so you don't overstay your welcome.
The UK is part of the EU but is not part of the Schengen area, a visa to the UK does not give access tot he Schengen area. That is why you had to get a UK visa not a Schengen one to enter the UK and why a UK visa has no use to you to go to Paris. As already stated above, you need a Schengen short-stay visa from France.
In fact, there are well over 100 countries that Americans with passports can travel to, visa-free. With just a valid American passport, you can travel to most countries in Europe, Central and South America, and the Caribbean, as well as many other popular tourist destinations.