Is there a tourist destination where the new year starts first and at the same time the old year ends last?

I am not sure this suits your question because you mention you want to be the first celebrating. Celebrating it twice can actually be easy to achive. You just need to find two countries (or places), next to each other, in different time zones. I would probably pick a place with two cities (which usually have more events), in each country, near the border. You could than quickly travel from one city to the other and still enjoy a bit of the old year (in the street or inside depending on your taste).

A good example for that is Spain and Portugal. You can first celebrate new year in Spain and than, one hour later, in Portugal. Here you even get the benefit of being inside schengen area so you don’t have to loose time showing identity/passport at the border. You can arrive more quickly to the next city.
Good examples of city’s in such condition are:

• Badajoz / Elvas (22 Km – Very nice cities with an old city center and old buildings)
• Berín / Chaves (30 Km – Very nice cities with tradition and very good food)
• Tui / Valença (2Km from center to center – smaller cities but you can even walk if you prefer)

Easily. Since Dec 29, 2011, Apia, Samoa lies in UTC+14; Pago Pago, Western Samoa is UTC-11, 25 hours — more than one entire day — behind. There are several flights a day between them on Inter Island Airways and Polynesian, taking all of 35 minutes.

So here’s a sample schedule. Party through to the new year in Apia, then take the first flight out at (say) 12 noon on January 1st Samoa time. You will land in Pago Pago at 11:35 AM Apia time, December 31st, so you can hit the beach, rest and recover, and then do it all over again. Woot!

Alternatively, if you really want to take it easy island style, you could even grab the 8-hour ferry between the two. Polynesians are not known for their strict adherence to published schedules though…

I’m not sure if this is quite what you asked for, but this press-release-based news story found its way onto a few news outlets (eg Yahoo News, Private Fly blog) in December 2013:

• Party at Sydney Opera House at 8pm local on December 31, till 1am local January 1

• Chauffeured to Sydney airport for a 2am local take off. This is UTC 1500 on 31/12.

• 13 hour flight to LA, arriving at UTC 0400 on 1/1, which is 8pm local on 31/12 because you crossed the International Date Line

• Another party at 8pm local on December 31; enjoy NYE again!

Cost per person: £9,157 (based on 15 passengers and a total charter cost of £137,355)

No.

Excluding Antarctica, the two closest points (by a large factor) on land which are on opposite sides of the date line are the Diomede Islands, which are about 4km apart. The International Date Line falls neatly between the western island (Big Diomede), which is part of Russia and uses UTC+12, and the eastern island (Little Diomede), which is in the US and uses UTC-9. These islands are sometimes known as “tomorrow island” and “yesterday island”.

Little Diomede has a little over 100 permanent inhabitants and can be reached by helicopter or by airplane in the winter. There are no formal hospitality structures — this isn’t a place where people go to party for the new year or otherwise. Big Diomede is home to a Russian military base and you cannot visit it under normal circumstances. In theory, I think you can walk between the two islands (December temperature still rarely go above -25°C = -12°F, so the ice shelf should still extend that far); however the Russian authorities are likely to object.

Anywhere else in the world, traveling across the IDL involves flying (or sailing, but it won’t be fast enough) between archipelagos. Likely journeys are Tonga (UTC+13) to Niue (UTC-11) (about 600km), Apia, Samoa (UTC+14) to Niue (UTC-11) (about 600km) or Kirimati, Kiribati (UTC+14) to Tahiti, French Polynesia (UTC-10) (about 2300km). I don’t know if there are any suitable scheduled flights.

Another option would be to sail on January 1 in a ship from a place that’s in the western hemisphere but west of the IDL, such as most of Kiribati: when you exit the territorial waters, the time switches from the local time of the adjoining land to the time determined by the longitude rounded to the nearest 15°. Conversely, you can arrange to be outside of territorial waters on December 1 and reach a place that’s in the eastern hemisphere but east of the IDL; this only includes US islands, none inhabited except for a few in the Aleutians.

The Island of Taveuni in Fiji has the international date line running right through it.

http://www.tripadvisor.com/LocationPhotoDirectLink-g297562-i22697088-Taveuni_Island.html

The earliest would be countries lying in UTC+14. In the Southern hemisphere summer, there are two places:

• Kiribati
• Samoa

The latest is UTC-12 in which only a couple of uninhabited islands lie. However in UTC-11 you might have better chances. One of the territories lying there is American Samoa. So if you manage to travel from Samoa to American Samoa you can have both the earliest and latest New Year lying merely apart 80 km. (from Upolu to Tutuila).

I’m not sure how easy this is however.

If you manage to travel there in less than one hour, you could theoretically even leave on January 1st and arrive on December 30th.