What is the best way to learn to speak Greek fluently?

There’s the generic answer: the fine old Greek saying, Η μισή ντροπή δική σου, η άλλη μισή δική τους. “Half the embarrassment is yours, the other half is theirs.”

Yes. They will think you sound ridiculous, no fear of that. They will also be hugely impressed (especially if they’re in the Greek diaspora), and will encourage your efforts. Dive in. And quote that proverb to them. Give ’em sass. They will love it.

Martin Pickering, back me up?

There are specific answers; see e.g. Dimitra Triantafyllidou’s answer to I’m learning Greek. What is the best way to improve my speaking/grammar skills?