Learn English – I wish and I’d rather

It can also be used to indicate that both options are negative, for example:

Would you like to go to a party?
I'd just as soon be eaten by wolves!

In this case, the respondent is using a situation which is clearly unpleasant (being eaten by wolves) to indicate that they don't want to go to a party. It should usually be easy to tell this is the intended meaning due to the hyperbole involved.

However, in your example:

I'd just as soon go home as go to a party.

The speaker may be indicating that whilst they don't necessarily want to go home, their preference (or lack of preference) for this is near-equal to that for going to a party. In this case, you would have to look at the wider context to understand the intended meaning.

I don't think I've ever heard it used differently to this (implying negativity), but this might just be unique to the British English I'm familiar with.

Perhaps you could give more complete examples.

"I'd like to ..." could be followed by a description of something that you would like to do. For example, "I'd like to swim the English Channel."

It could also be a response to a question about things that you have done, to express the idea that you have not done it but want to. "Have you ever swum the English Channel?" "No, but I'd like to." Or it could be a positive response to an invitation. "Perhaps you and I could swim the English Channel tomorrow?" "Oh, I'd like to."

"I'd like that." is a complete sentence. It means essentially the same thing as my last example above. "Perhaps you and I could swim the English Channel tomorrow?" "Oh, I'd like that."