The first two sentences mean pretty much the same although the second is more conversational. Both sentences require care with intonation or otherwise can sound bossy and rude. The second two sentences are also correct. They are slightly different. Both suggest a way of doing something but the second one is much stronger. If the person spoken to does something different, they have ignored the speaker's explicit preference.

As the others have said, the first two sentences are about the same in meaning. "you should" is more formal ("you'd better" is a colloquialism), so in any formal setting (essay, email, business), I'd recommend using you should. "You'd better" can also be used in a threatening way, e.g. "You'd better not come back here, or I'll break your neck." In American English, rather = prefer. So the difference between these two sentences is: I would (be willing to) use this way, but I would rather (prefer to) use a different way. In British English, it is common to use "rather" to offer an alternative. E.g. "You could take this route, or rather, you could take this shortcut." (Rather can also mean somewhat: "I'm rather tired." But in general, neither of the sentences you've listed sound very natural in English. If by "way" you mean "method," you should say, "I'd do it this way." / "I'd rather do it this way." If by "way" you mean "route/road," you should say, "I'd take this route."/ "I'd rather take this route." :)

We use the word "rather"to out emphasis on sth also make the statement with greater degree so "I'd rather use this way"is much stronger than the "I'd use this way".the first two sentences are equally the same "should=had better"