Your achievements are tied to your gamertag, not the console, so there are a few ways to transfer everything over.
If you have a hard drive on your old console and you're buying one of the older models, you can just pop it over to the new one and your achievements and saved games will be usable right away. If you are buying one of the newer consoles with internal hard drives, you should use a transfer cable to move all the data over to the new console:
If you've purchased a new Xbox 360 console, you can transfer everything (games, videos, music, saved games, gamertags, and even your hard-won achievements) to your new console!
Two options are available for moving your precious information. You can either use the Xbox 360 Hard Drive Transfer Cable to move everything to your new console, or you can connect a USB flash drive if you have only a small amount of content such as your gamertag and saved games.
If you don't have a hard drive, you can either move the data over via the USB memory stick or do a recovery on your gamertag. The recovery will not transfer any hard drive data and will only retrieve your profile. Also, if you do a recovery, be aware that every achievement you unlocked since the last time you connected to Xbox Live will be lost.
From the support page:
You might need to recover an Xbox
LIVE gamertag in the following situations:
- You have a Zune or Games
for Windows — LIVE account and want to join Xbox LIVE.
accidentally deleted your profile, or you want to move your profile to
a new console.
- Any achievements or changes that you have
made since the account last connected to Xbox LIVE will be lost during
the recovery process.
- Do not recover your gamertag to a friend's
console as a convenience. You might lose your saved games. Instead,
follow the steps in Use a gamer profile on another console.
If you have any downloadable content, you might also have to transfer content licenses to your new console.