Japanese is really easy to pronounce. Every vowel is always the same no matter where in the word it comes.
a - ah, rhymes with "cot"
i - ee, rhymes with "tea"
u - oo, rhymes with "two"
e - eh, rhymes with "meh"
o - oh, rhymes with "toe"
That's it. As long as you stick to those sounds, and don't try to use any American vowel sounds like the short a in "cat" or the long i in "kite," (they don't exist), you're good.
And stick to them
no matter where they come in the word: sake is pronounced sah-keh, not sah-kee or seyk.
Westerners almost always get these wrong because they're used to vowels that change pronunciation depending where they are in the word. Don't do that!
So by that rule,
Hagun = "hah-goon"
Nanatsusaya = "nah-nah-tsoo-sah-yah"
Hachiman = "hah-chee-mahn"
Seiryu = "seh-ee-ryoo" (When spoken quickly, the eh and ee here blend into something that sounds like "say-ryoo")