Can I transfer my contacts from an old non-iphone to my new iphone?

I already activated the iphone which means I don't think I can get my contact information from my old phone due to it not having service any longer. Aren't the sim cards different sizes? And if they are, how can I make the transfer?

Which phone is that? If it is an android phone, there are some simple and effective ways to transfer contacts. This can help you get started
http://techpp.com/2011/11/24/how-to-transfer-contacts-from-iphone-to-android-and-vice-versa/