Like for a word like はい。 Are you supposed to sound out the あ and い separately or together? Because the way I'm hearing it, it sounds like you say it together.

Thanks.

You assumed correct, it's together.

They're actually two separate syllables that are said separately but often run together to make it sound like one syllable. It's the same as the ん syllable. Sometimes people say "たぶん" like the ぶ and ん is one syllable, but sometimes people will pronounce the extra syllable and say た　ぶ　ん。

As in most languages, combinations of vowels are often sounded together. This comes from a natural blend of the sounds when spoken quickly. In spoken language, the vowels aren't normally spoken seperately, but sometimes you'll hear some people do that, usually to draw out the word and place more emphasis on it. It's much more common in music to pronounce the vowels individually, because that way the word can fit alongside the music. So in songs you'll often hear that, but in conversation not so much.

It's like the English word "Aye" but with a H.

It doesn't matter too much whether you blend the /a/ and /i/ vowels together or pronounce them as two steps, as long as together they take two units of time. In 汚い（きたない） 「きた」 should take as much time as 「ない」.

Most importantly, just keep listening, and say it like it sounds.