If you need a visa to go to your host family then make sure you get that visa before you buy the ticket otherwise you’ll loose some money in case you don’t get it. This is just a precaution. For example if you live in Poland and your host family is in Germany just across the border in case you don’t get your visa you’ll loose much less money than from US to Australia. (These countries are an example).
As an advice, get your visa two weeks before the date of your departure – in case it goes wrong and you don’t get your visa the family could find somebody else.
In any case you need a letter of invitation from the family. Do not go to a family before you get this letter. In this letter are the details about your job. Ask your host family to include your duties in that letter. The invitation letter can be sent to you by post, by fax, or by email (the one you had your conversation with them)
If you go trough an Au Pair agency then they take care of all this aspects but if you found each other over the internet and you need a visa to get to them you might need some other document. You need a copy of their passport and a copy of their bank statement or any other document proving that they don’t have financial difficulties. This is usually required by the embassy. Better is to phone or email the embassy and ask.
After you are allowed you can go to meet the family. The travel costs are paid by you.
Usually the host family picks you up from airport or train station. You have to speak to them about these details.