I've a '98 SI with the original wheels and 195/55 15 tyres on. My question is if I wanted to change to 16" wheels what size tyres would I need? Do I need to make sure I get the same width wheels as my current ones too?

id get 205/45/16`s

I thought standard 15's were 195 50 15

Yes, you're right they are.

id get 205/45/16`s

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215/40x16 is a better option imo