The only time that I would use the word "regretful" would be in a sentence such as "I found his choice regretful."

I agree with Lisa: say "I regret my choice".

In American English, what Alexandra suggests would not sound right even though it was grammatically correct. We would say instead, I regret my choice. [at the time] I regretted my choce [but now ...] For Tania and her exam, we would say "I was sorry for Tania since she failed her exam." Maybe it is just our local (California) usage, but I've never heard regretful used in either of these usages in real life.

In this sentence, you would use "of". You would not always use "of" with "regretful"; it depends. For instance, if you were writing about feeling regret for a friend, you might say "I was regretful for Tania, since she failed her exam." In this sentence, if you said "I was regretful of Tania" it would imply that you did not like Tania, and you regretted meeting her. In your sentence, you did not like your choice, so you are regretful "of" your choice.