The best option is to take the train from Moscow to Abakan (76 hours), with one-way fares varying from approximately 4,000 Rub (US$160) to 9,500 Rub (US$390). The price changes according to peak tourist seasons in Russia: by far the cheapest day to travel is the 1st of January, while July is the most expensive.
It is possible to take a bus from Abakan in neighboring Khakassia. While a night bus is an option, don't take it! The route meanders through the gorgeous Sayan mountain landscapes of the Ergaki region, which would be foolish to miss. The train from Moscow arrives in Abakan at 6 AM, and apart from the bus, there will be plenty of taxi drivers offering to take you to Kyzyl (approximately 420 km from Abakan) for 1,000 - 1,500 Rub (US$40-60). There is a possibility that you may have to register at the Russian-Tuvan border - just hand over your passport to the police officer, there is no fee for the service.
Kyzyl Airport is small and offers flights to and from the Siberian center of Krasnoyarsk as well as Raduzhny, Khantia-Mansia for some odd reason. The direct flights from Moscow to Kyzyl have been resumed as of Fall 2007, the flight goes twice a week via Ufa, Bashkiria's capital, departing from Moscow on Sunday and Wednesday.
The town is relatively small, so if you are staying in one of the hotels in the centre, all of the landmarks will be within walking distance. There are no buses in Kyzyl, but there are plenty of route taxi vans ('marshrutka') and regular taxis. The cost of one ride on the marshrutka in 2008 is 14 Rub (approx. US$0.57), regardless of the distance.
The Centre of Asia Monument - located on the picturesque bank of the Yenisey River, where the Small and Big Yenisey Rivers join into the Great Yenisey. It is quite easy to find - just walk north from Hotel Kyzyl until you hit the river bank, then turn right and keep walking until you see it.
Shaman centre - located to the left of the Centre of Asia Monument. For a small fee, you can have your fortune told or an illness cured by a Tuvan Shaman. In summer, the government puts up a decorative yurt (traditional nomadic house) next to the centre.
National Theatre - located on the main city square opposite the White House (Government Building). Note the wooden carvings on the building, which were hand-crafted in the traditional Scythian 'animal' style.
Lenin monument - located on the main square, just next to the National Theatre.
National Museum - located about 500 metres west from the National Theatre, down the main road. The new building opened in 2008, and some exhibits have not yet been moved from the old building. It features exhibits on the Tuvan flora and fauna, religion and traditions, archaeological findings from the Stone Age and the Scythian era, as well as exhibits from WWII and the Tannu Tuva Independent Republic. Surprisingly, the Museum also features paintings of renowned Russian and foreign artists, including Shishkin and Reynolds. Entry costs 100 Rub (US$4) for Russians, and 200 Rub (US$8) for foreign tourists. The Museum is open until 6 pm, and is closed on Mondays.
National Theatre - In summer, the Theatre hosts different festivals (traditional dance, throat singing etc.) so keep an eye out for posters and advertisments around town. Tickets cost around 100 - 300 Rub (US$4-12).