While there is no "rule", the logic behind making a connection with someone new - who is obviously not on your speed dial, is to do it as least intrusively as possible. First you gather as much contact information as possible. If you can only find one type of contact information - say a Linkedin introduction request, then you do that. Easy, since there aren't a lot of options, but if you have plenty of information, the least intrusive is best, namely email. Email serves as a good introduction platform, since you can say as much about yourself and your purposing in contacting someone, so that when they go back to it, to contact you, they have the necessary information about you and correct prompts and reminders about the purpose of your business together. The next line of contact is a business phone number, then any other phone number and really for professional purposes you should not really text someone initially, since it's intrusive, and easy to ignore. For personal business, it might be equally ignored and intrusive, but hey, that one is up to you.
The latter question about how to contact someone back has an easy response, but is a rule that seems really difficult for people to follow. Both in business life and in personal life, if people can learn and share this rule, communications will be improved exponentially. The rule of thumb to contact people back is to do it in the same medium they contacted you, unless stated otherwise. If you receive an email, and business is left pending, you will email them back every time, unless stated otherwise, "after you read this, please give me a call", or "when you have that information, please skype it to me," or "oh yes, I know them, I'll introduce you to Joe Smith on LinkedIn."
Why is that the case? They say effective communication has to do with the ability to "meet people where they are at," often this is meant in terms of outlook and perspective, but first of all you need to do so pragmatically. If they emailed you it's probably because that's either the most effective or convenient form of communication for them, so in order to establish a good starting point, meeting the other party on the same platform is key. Besides, it's in your best interest that they are receptive to you and paying attention. It behooves you to be aware of people's capabilities and have reasonable expectations and adapt when necessary. In my personal life, I do not enjoy talking on the phone, which does not at all mean I don't absolutely enjoy time well spent with family and friend and talking with them face to face, yet I'll try to schedule time with them usually via a short, short text. "Lunch tomorrow? where?" However, my grandmother doesn't text and my family members in other countries don't text and I can't meet them, so I make an effort to call them. Sure I'm not a fan of the phone, but it's all they can do, and if I don't adapt, then I lose out. Right?
Is it possible to switch platforms to something more convenient for YOU rather than the way you were first contacted? Sure, just say so in a reply in the original platform. I do it all the time! "@thedigitalinc, what's your email address? I'll send you a longer message shortly" or "I'm going to be running around today, it's easier for me to call you, what's the best number and time to reach you at?"
We live in days where come 8 AM almost every morning, we invariably get inundated with emails, Skype messages, Facebook messages, pop up chats in Email, and Twitter replies; how else are we going to be able to keep the communication flowing if we don't follow some sort of communication rules?