Can a person fly to the Moon as a tourist?

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.

Another more graphical answer:

With the privatisation of space travel business, there are some interesting projects in pipeline.
Check the below links:

1. Virgin Galactic - Richard Branson's Virgin Group Venture
2. Blue Origin - Jeff Bezos' (Amazon CEO) venture
3. Space Elevator
4. List of private spaceflight companies

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.

Wikipedia has a page on that topic: Tourism on the Moon:

Space tourism companies which have announced they are pursuing lunar tourism include Space Adventures,[4] Excalibur Almaz,[4] Virgin Galactic[2] and SpaceX.[3]

The SpaceX project, dearMoon project, is one of the most likely projects to be successful in my opinion, but is not planned to take place before 2023.