Ras Al Khaimah is the northernmost of the seven emirates, and the one that looks least like the Dubai on your screen. Its name means roughly "headland of the tent", and the Al Qasimi family has ruled it since 1708. Behind the coast the Hajar Mountains rise straight out of the plain. The tourism board calls Jebel Jais the UAE's highest peak at 1,934 m, though the true summit sits just across the Omani line and the highest point on the UAE side is about 1,892 m. Either way it's the one place in the country where you pack a jacket. The medieval port of Julfar stood here until the Portuguese arrived, and in December 1819 British forces took the hill fort at Dhayah, which led to the 1820 treaties that shaped the future UAE.
This is a nature and beach guide, not a city one. The city is a working port with a creek and a Corniche, and nobody comes for its skyline. They come for a 2.83 km zipline over a canyon, forts and villages that haven't been turned into theme parks, and long beaches on Al Marjan Island and in Al Hamra. Three days is the right length: the mountain needs a full day, the coast and its heritage a second, and the third goes to a dhow trip across the border into Oman's Musandam fjords.
Day one is Jebel Jais, day two the forts, the ghost village and the beach, day three the Musandam trip, the only one that needs your passport. Start on a Wednesday: the Jais Flight and the Sky Tour run Wednesday to Sunday and the National Museum is shut on Mondays. Ras Al Khaimah is a little over an hour from Dubai by road, and the United Arab Emirates page covers the rest of the country.