'Much happier' is correct. We only use 'more' with adjectives which use 'more' in the comparative form. The comparative of happy is happier. So 'much happier'. The comparative of 'expensive' is 'more expensive'. So 'much more expensive'. You can also use 'a lot'. A lot happier, a lot more expensive.

Much happier

This is not a 100% rule, but usually if a word has only one or two syllables (like "happy") we add -er to make the comparative and -est to make the superlative ("happier" and "happiest"). When a word has three or more syllables, we use "more" and "most." We never use both -er + more or -est + most.