There are several clubs in Prague, four of them in the premier division, all with experience with foreign players. They are Slavia, Sparta, Praga and Tatra. There are about four in the second division, worth mentioning is especially Prague Barbarians, who consist mostly of British and French guys working in Prague.
Dragon is the choice if you want to stay in Brno. Good club and very friendly environment.
See short stay. Check whether you need visa: US, Europe is visa-free, Australians and NZ need visa. Former Soviet republics need visa as well. Beware: Canadians need visa as well (written in 2002)
It is quite easy to get a job in Prague, with an average salary, depending on skill and luck. The bad point is accomodation. While the salaries are low (300 GBP a month), rents are high (150GBP for 1 bedroomed flat). The best thing is to select a club and get in touch with them. It is quite common they will come up with an arrangement. Don't worry about the work permit.
The standards are similar to the county club level rugby in Britain or 3rd division in France. The fitness of some of the guys is impressive, though. Practises are usually held twice a week, Tuesdays and Thursdays. Season runs in two parts, Autumn (August-November) and Spring (March-June). Spring in Czech is really wonderful.
There is plenty to do in Prague or Brno. Just check the Internet - there are concerts, sports, films (in English), satellite TV, lots of pubs (American, English, Irish).
If you want to get more information, please contact the Prague Rugby Travel guys at PragueRugbyTours.com, who are official tour operators of the Czech Rugby Union.