At the airport: passport control
If there is a trainee sign displayed in the counter, better choose another one.
By law, you must carry your passport with you at all times
More reputable agencies require that valid licences be produced: foreigners who do not have a Thai driving licence must carry a valid International Driving Permit. Even if you manage to rent a car without an IDP, not having one will invalidate the insurance and count against you in the event of an accident.
ATMs can be found in all cities and large towns, and international withdrawals are not a problem. When using a debit card, an ATM will typically provide a much better exchange rate than a money exchange counter, and this is especially the case if you have a card that does not charge a transaction fee for overseas withdrawals (becoming common in countries such as Australia). ATMs are available at Bangkok's Suvarnabhumi airport (BKK) after collecting your bag and clearing customs, and while it is advisable to arrive with a small amount of baht if possible, you may obtain cash from an ATM after landing as well. Since May 2009, there is a 150 baht surcharge for use of foreign ATM cards on almost all banks ATMs. The banks which, as of June 2009, do not charge this (quite unfair to the foreigners) new fee are mostly foreign-based ones - UOB, HSBC and Aeon at least (also, there are mixed reports about Thailand's GSB). Anyway, you'll be notified about this fee in any ATM which charges it before commiting transaction - so you always have an option to cancel.