[Note that this answer is specifically for American English]
Yes. There are differences. First of all, the first one is incorrect.
"I'd sooner we travelled by plane" doesn't really mean anything. My guess is that you got something confused with this construction:
I'd sooner eat a bowl of nails than eat another bowl of your mother's soup.
This is a way of saying (very strongly) that you won't do something. So you choose something that is something that you obviously won't do because it is so extreme (eat a bowl of nails) and say that you'd sooner do that than whatever it is you're trying to say that you don't want to do.
The other two are very similar in most cases, but not in all.
b.— I'd rather we travelled by plane.
c.— I'd prefer we travelled by plane.
The place where they differ is in this case:
Person A: How would you like to travel?
The answer to this question can't be "I'd rather we travelled by plane." You'd have to use the other option
Person B: I'd prefer we travelled by plane.
If you're using rather, you're replying to a suggestion...
Person A: How about we travel by boat?
Person B: I'd rather we travelled by plane.
However, in this case, you could also use "I'd prefer."