The short answer: slowly.
I’ve been there. Believe me. You get a request. Or you decide that a certain publishing house is PERFECT for your book. OMGOMGOMGOMGOMGOMGOMG… you have to send it there right now because it should’ve already been there three weeks ago with how anxious you’re feeling so you run out to the post office and you shout “Overnight it!” and then you whip out your debit card and and and…
Let me repeat: I’ve been there. This sort of excitement — a request from an agent, an editor you meet and adore at a conference — can inspire some reckless, I’m-in-love spending.
But what you should really be spending is the time to perfect your manuscript. Publishing is notoriously slow. Unless the agent or editor says, explicitly, “Get this to me ASAP” and for some reason you can’t e-mail it to them, don’t waste your money Overnighting, Expressing or Prioritizing anything.
Here at the publisher where I work, we sometimes get unsolicited submissions overnighted to us. How much did it cost to send those five pages? I don’t want to know. If you are sending in a regular submission to an agent or editor, this is what will happen to it: it will arrive, it will be sorted by the mailroom, it will sit in a bin, it will sit in a bin some more, an intern will glance at it, it might sit in a bin again, someone might recommend it to an editor, it will sit on the editor’s desk, the editor might glance at it, it will sit on the desk some more… It is a sloooow process.
I repeat: the only time you should make haste sending anything is a) when your project is absolutely ready for consumption and b) when the agent or editor explicitly requests that the thing is sent to them in an expedited fashion.
Otherwise, good old first class cheap-o mail is fine. It is encouraged, even. There are few things sadder than watching an Overnight package languish in slush for a month. Don’t be That Writer.
You’ve got a fire under your butt and you’re excited. Good. Express that fire by writing, revising or otherwise improving your craft. Don’t let it rocket-boost you to the post office. There are much better ways to channel all that energy.