Currently the only variant to visit the cosmos is to go to the International Space Station
The only way to visit the ISS is to go there aboard a Russian Soyuz (spacecraft) .
Note: you must go through some training before this.
The price is up to $40 000 000 (and $15 000 000 more to go to open space).
In the near future the Russian company RosCosmos is going to provide tours around the Moon, but it is still a project, not a real tour perspective.
Other tours are all about near-Earth flights.
Update Space Adventures announced the first commercial flight to the Moon in 2017 (50 years from Appolo 1 flight), the price is $300.000.000 for two places.
Flight will be done via russian Soyuz.
Good luck!
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Unfortunately since 1972, travel to the moon has been non-existent, tourism or business.
Currently the best bets to get yourself to the moon are with the USA, the European Space Agency and China - all have serious space programs with intentions to return, but are in no hurry - indeed the European Space Agency prefers robotic missions in general.
NASA was targeting their next trip for 2018, but that now looks to be perhaps a decade later than that.
China is aiming for 2024, and India is perhaps dreaming but claims to want to go in 2020.
Naturally places are limited on these crafts so you'll want to get in early.
Prices are difficult to estimate.
Space tourism to the International Space Station is generally on the Soyuz rockets which can only reach Low Earth Orbit, and trips are generally believed to be around US$20 million per person.
If you're content with just a look at the moon, the private company Space Adventures and the Russian Space Agency have floated the idea of a flight around the moon for around US$100 million.
Getting around is more difficult once there, you can't fly as there's no atmosphere, and unless you bring a rover/vehicle you're stuck walking.
If you can manage to land on Mons Hadley, the Descartes Highland or the Taurus-Littrow valley there are three stranded moon buggies from previous trips that may have some power left in them for your purposes.
There's not much to do once there - but the view of Earth is outstanding.
You can play golf - although there's no official courses, but with the low gravity you'll set a personal best for driving, I promise!
And of course, where better a place to do the moonwalk than on the Sea of Tranquillity, site of the original manned moon landing!
Don't forget to bring food and drink with you, as there are currently no markets or shops on the surfaces, although Dominos is planning on opening a branch of its pizza chain there soon.
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Another more graphical answer:
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With the privatisation of space travel business, there are some interesting projects in pipeline.
Check the below links:
Virgin Galactic - Richard Branson's Virgin Group Venture Blue Origin - Jeff Bezos' (Amazon CEO) venture Space Elevator List of private spaceflight companies
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A company, Golden Spike , offers bookings for lunar tours in the next decade.
According to news articles , the fee is as low as $1.4 billion.
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I don't want to really fly there, it would take too long and probably would be prohibitively expensive for me.  But nevertheless I am really wondering if it is already possible, and if so, how expensive it is and which companies might offer such ventures?
Can a person fly to the Moon as a tourist?
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<adventure><long-haul><extreme-tourism><remote-locations><space>
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Nobody has travelled to the moon since 1972 for any reason, but the private company Space Adventures and the Russian Space Agency have floated the idea of a flight around the moon for around US$100 million. Commercial trips to the International Space Station are possible using the Soyuz rockets, which ar ecapable of reaching low earth orbit, and are generally believed to be around US$20 million per person. Another company,  Golden Spike , offers bookings for lunar tours in the next decade.   Although nobody has travelled to the moon since 1972, some companies are offering trips into space in the future. Prices range from 20 million to 3 billion dollars.
A number of projects appear to be offering trips into space in the near future at prices ranging from 20 million to 3 billion dollars. Since 1972, travel to the moon has been non-existent.