How formal do you have to be when you're starting a professional relationship with a new supplier  or customer?

I would definitely avoid stiff and boring, but at the same time keep a formal tone to your email or phone conversation. 

Most businessmen and women from Great Britain or the US are probably not using the surname in the conversations anymore when they've spoken to the person a couple of times, or have had a few emails going back and forth. 

However, I would suggest starting by calling them Ms Smith or Mr Jones and ending your correspondence with "Best regards" and your full name. 

When you call the next time, you can ask to speak to Mr Jones and gauge his response. If he greets you with your first name, you can stick to first names from now on. 

When visiting your new acquaintance at his/her office, ask for their  full name and present yourself with your full name to the receptionist. If you by this point are on first name basis with your contact, you should only address them with their given name when you get to meet them personally. If you are not on first name basis yet, you should stick to full name approach until they show you it is ok to drop the titles. 

Most countries greet each business contact with a handshake, stretching out your right hand and shaking your contact's right hand for a second or two. 

But there are of course, exceptions to this rule. The custom in some countries is bow to one another and keep the palms of your hands together in front of your chest. 

Some people kiss each other on the cheeks and the number of kisses differs from country to country. 

Check this cultural etiquette out carefully before you are due to meet your contact, to avoid an awkward moment when greeting each other.