Yes, you can keep your old apps when you get a new phone...if you know the tricks.
Even if you are not intending to get a new phone, use
My Backup Pro and
AppBrain to make sure that you don't lose everything if you lose, drown, drop, or otherwise toast your phone.
Before you get the new phone use
My Backup Pro to backup all apps and data to their website and create zip files for your SD card. (Must be Pro version, $3.) My Backup Pro will automatically backup your apps and data every day so that you have a current backup if trouble strikes. Do a manual backup to the SD card now and then copy these zip files to your computer. The My Backup Pro data backup zip file includes Contacts, Text Messages, Call log, Bookmarks, Calendar, Home Shortcuts, Alarms, Dictionary, etc. That's just as important as your apps.
ASTRO Manager and
SystemPanel Pro ($3) will also let you manually backup apps but
will not make automatic daily backups or backup data or backup to a server.
Titanium Backup can backup apps and data of rooted phones.
Also before you get the new phone, use
AppBrain to create a list of your apps. When you get the new phone, AppBrain should let you automatically install all of your old apps on the new phone.
On the new phone, use the same Gmail account login and password. Most of your contacts, calendar, and certain paid apps should come to your new Android automatically. But note that many of our favorite paid apps are not paid through Android Marketplace but directly through the developer and will not transfer.
Install
AppBrain on the new phone and click "Manage my apps" and then "Synch with AppBrain" to copy all apps to the new phone.
Let
My Backup Pro Restore your contacts, past Messages, etc. I would NOT restore Settings as at best they will not work on the new phone and at worst could cause problems. No two Androids have the same setting options. You can also use the zip backup it created to reinstall a missing app but that should not be needed.
It is never painless to go to a new phone but AppBrain and My Backup Pro can lessen the blow.