1. Without connecting your iPad to your MacBook Air -
A) Install iTunes / make sure the latest version of iTunes is installed.
B) Go to the Store tab and click on "Authorise This Computer".
C) Go to Edit, click on Preferences and click on Devices. Put a check mark in the box next to "Prevent iPods/iPhones/iPads from syncing automatically". Click OK and close iTunes.
2. Connect your iPad to your MacBook Air -
A) Start iTunes (if it does not do so automatically).
B) Right click on your iPad under Devices in the left column. Select "Transfer purchases".
C) When the transfer has completed, right click on your iPad again and select "Backup".
D) When the backup has completed, left click on your iPad, click on the Apps tab, and put a check mark in the box next to "Sync Apps", and then click on "Apply" (bottom right).
E) Unless you want to always do a manual sync (as I do), go back to Edit, Preferences and uncheck the box next to "Prevent your iPod.....".
F) Finally, on your iPad, check that everything is syncing properly. If not, go back to iTunes and check all settings.
NB You should not lose any apps. However, any 'purchased' missing apps can be downloaded again FOC. As with updates, this must be done with the Apple ID under which they were originally 'purchased'.
Also see here as to moving your music to your new computer -
The absolute best way to do this would be to copy then iTunes library from the old Mac to the new MBA. What Alan is describing will work if you cannot copy the library to the new Mac.
One little correction - when you launch iTunes on the Mac - go to iTunes>Preferences>Devices to turn off auto sync. Edit>Preferences>Devices is for iTunes in the Windows platform.