Formality?

At the gym where I practice budo I heared some guys talking about the japanese way to thank someone, and the formality used with that. this is what they were saying:

Domo arigato gozaimasu: really formal
Domo arigato: formal
Arigato gozaimasu: formal
Arigato: common

But I don't really believe it directly, so I thought it would be worth asking if they were right.
If possible, I would like to know the most common ways to thank someone, so also the most common way for saying it formal, really formal, rude, etc.


after the class is over we say "domo arigato gozaimasu" to thank the teacher for the lessons, is there also something that is being said in japan when they start the classes?

I practice karate, and as far as I know they say "arigatou gozaimashita" in karate dojos in Japan, which is past tense formal. As for politeness generally, I would say "arigatou gozaimasu" is what you'd say if you for example recieve a gift from your boss (or training from your teacher, as in karate), "arigatou" if you recieve a gift from a friend, and "doumo" if your friend hands you the table salt. In general, the longer, the more polite it is.

EDIT: Sorry, I missed your last question. At the beginning of lessons, the common phrase is "onegai shimasu", which is like a polite way of requesting something.

duomo arigatou gozaimasu ... very polite and formal "thank you very much"
arigatou gozaimasu ... polite "thank you"
doumo arigatou ... casual "thank you"
arigatou ... casual "thanks"
doumo ... very casual "ta"

Before the class, you can say "yoroshiku onegai shimasu".

As hiptanaka says, "gozaimashita" would fit better after the class/lesson is finished.

okay

I think "arigatou gozaimasu" is the one I need now ^_^