How can I easily deny other apps from accessing my personal data? If I use the standard Android contacts, sms and phone app all these data are endangered to be read by other apps.

Is there a solution without rooting the phone that every simple user could easily install?

Sure,you could conceal your contacts by renaming them to something cryptic. But that is not very handy.

Maybe there are alternative apps for contacts, sms and phone app? So there would be no data in the standard apps.

Or some simple solution that prevents access?
Short answer with two letters: No.
TL;DR
Whatever app you use to work with your contacts, SMS, and phone – they will need to access your contacts, SMS, and phone (how else should that work?), so alternative apps for that won't be much help. With the exception of those few which use own databases (I know of some calendar apps that do this, but did not yet encounter any such contacts/dialer/SMS app), which would not work with contacts (how shall the dialer or messages app then access them, except it's built into the very same app?).
Let's assume you find such an all-in-one contacts app using its own storage: you had to trust it as well that it doesn't initiate calls to some premium numbers (or send SMS there) in the background, and (in case it has the INTERNET permission) send all your private data &quot;to some cloud&quot;.
So as Dan pointed out in his comment on your question: If you have concerns about an app that asks for permission to access something, don't install it.
With root
If, on the other hand, you decide to root your device, there are several solutions available to restrict apps. Just to name some: LBE, PDroid, XPrivacy. for details, please check with What information does stock Android send to Google by default, and how do I opt-out? and how to fake my personal information.