Learn Polish: Dziękuję bardzo. - Thank you very much.

Yes! Unlike English where we just say "thank you", in Polish you use the verb dziękować (to thank). Because it ends in -ę, it is specifically the "I" form. So, dziękuję literally means "I thank".

How do we get from the dictionary form dziękować to dziękuję?

This belongs to the special -ować verb group you are learning now. To conjugate it in the present tense, you drop the -ować ending and replace it with -uj-. Then you add your standard endings. For "I", you add -ę, giving you dzięk-uj-ę.

If dziękuję means "I thank", how do I say "he thanks" or "you thank"?

You use the same -uj- stem and add the regular endings for this verb group. "You thank" is dziękujesz (adding -esz) and "he/she thanks" is dziękuje (adding -e). Just be careful: only the "I" form (dziękuję) is used as a phrase to say "thank you" to someone!

Can bardzo be used in other situations, or is it only for "very much" here?

It is a very common word meaning "very" or "very much". You can use it to intensify adjectives, like bardzo dobry (very good), or with other verbs, like bardzo lubię (I like very much).