Can I transfer all my apps from my old Iphone to my new Iphone?

Yes. Sync the old phone to itunes. Activate the new phone and after you have activated it you will be asked if you want to restore (or whatever apple call it) the phone from the back up.

After itunes is finished all the settings, mail accounts, text messages etc will be on the new phone along with your apps.

With the backed-up “old” iPhone you will get with the iTunes sync an exact software-copy

… if it is not jailbroken with cydia-apps… these apps are not incl. in the sync

As long as you have them on iTunes and they are compatible then you should have no problem.