I'd like to supplement Boaz's excellent answer with some concrete examples, for those members who learn better by example.
Using this categorization system, you can say that linking particles may generally follow grammatical role particle when they indicate a role in a sentence...
僕に難しすぎるよ。It is too difficult for me.
- 僕には難しすぎるよ。[Others may find it easy, but] it is too difficult for me.
- 僕にも難しすぎるよ。It is too difficult for me as well.
- 僕にさえ難しすぎるよ。It is too difficult even for me.
...が and を has a strange property: when you add most linking particles to them, they disappear and are replaced entirely by は、も or whatever linking particle you've added, and you can no longer differentiate between subjects and direct objects.
猫が犬を噛んだ。The cat bit the dog.
- x 猫もが犬をさえ噛んだ。
- 猫も犬を噛んだ。The cat bit the dog too [after the dog was bitten by something else].
- 猫が犬さえ噛んだ。The cat bit even the dog.
- 猫は犬しか噛まない。The cat bites nothing but dogs.
Sentence-final particles obviously can't follow grammatical role particles or linking particles, since particles of these classes only come after the parts of the sentence that appear before the verb, while sentence particles only follow verbs.
その本を俺に渡せ。Hand that book over to me.
- x その本をぞ俺に渡せ。
- x その本を俺にな渡せ。
They usually intermingle only with other sentence-final particles (or with interjection particles that double as sentence-final such as さ and ね).
それは嘘です。That's a lie.
- それは嘘ですよね。[I'm pretty sure] that's a lie [but I'm asking to make sure].
- それは嘘かよ。That can't be lie! [asserting disbelief]
Connective particles also don't mix very well, since they usually come in the middle of a list of nouns, where no other article (even an interjection particle) can be placed. They also don't tend to mix with each other.
サツキとメイは姉妹です。Satsuki and Mei are sisters.
- x サツキとだけメイは姉妹です。(It makes no sense to qualify the と because it applies equally to both)
The interjection particle ね can generally follow almost any other particle, and really is very flexible, while the vocative よ usually follows only nouns or names. I think さ is somewhere in the middle.
僕は姉と映画を見にいった。I went to see a movie with my sister.
- 僕はね、姉とね、映画を見にいった。I, um! Went to see a movie! With my sister! (five-year-old boy)
- 僕はさ、姉とさ、映画を見にいった。I like, went to see a movie with, like, my sister. (thirteen-year-old boy)
I would add that adverbial particles can link up with grammatical role particles and linking particles.
僕だけが分かる。 Only I understand.
それほどは悲しくないぞ。 I'm not that sad.
リスなどもいる。[In addition to other things,] there are squirrels and other creatures like that.