My iPad is showing no iOS update. But I can't update some of my apps because it says it's outdated. Any ideas?

What version do you have (Settings > General > About > Version) ? If you are on iOS 5+ then you should have a Settings > General > Software Update option for direct update, if you have iOS 4 then you will need to do the update via your computer's iTunes (which needs to be failry up-to-date).

To update via your computer, connect the iPad to your computer's iTunes and copy any purchases off the iPad to your computer via File > Transfer Purchases. You may also want to copy photos and any important documents off the iPad as well e.g. via the file sharing section at the bottom of the device's apps tab when connected to iTunes, via wifi, email, dropbox etc - they should be included in the backup, but it's best to have a copy of them outside of the backup just in case. You can then force a backup of the iPad by right-clicking the iPad 'Device' on the left-hand side of iTunes and selecting 'Backup'.

Then start the update by selecting the iPad on the left-hand side of iTunes, and on the Summary tab on the right-hand side clicking the Check For Updates button

Updating to iOS 5+ : http://support.apple.com/kb/HT4972

A first gen iPad should update to iOS 5.1.1 and an iPad 2 and 3 to 6.0.1 (which is the only version that is 'signed' by Apple) - with iOS 6 will lose the built-in YouTube app and the Maps app will change.