I am in the process of moving my web/mail server to a new server and putting them on new IP's (different hosting ISP).
I want to change the direction of my http client and smtp client requests from hitting the old one to the new server,
so in this case which zone files do i need to request from my old ISP, to prepare zone templates on my new server (new ISP)
to avoid surfers from going to old server/IP.
My domain name will not change , the redirection will require only to change the IP.

Are you actually moving your DNS provider as well, or just the server?

In either case unless you have a large number of entries you don't need to look at the old one, just provide the new details to whoever provides your DNS.

The information that is needed are the MX entries (which need to give fully qualified domain names for your mail server), and A records (e.g. for www, and if email is provided on your own server rather than your ISPs mail server a mail A record) with the IP address of your new web server.

The rest of the details (SOA header etc.) will be created by your DNS provider.

You can check your current details using the dig command

(will show current mail settings)

(will show A record)
- repeat for any other subdomains
e.g.
dig a mail.yourdomain.com (if using own mail server)

I am moving also my DNS.

The new DNS provider will need to know:

Mail Servers (and priorities, if more than one)

and

Any A records (e.g. any Name to address translations).

and any CNAMES (e.g. aliases).

For a standard web server (using someone else to handle your mail), then you would provide them with

Mydomainname.com
----------------

MX - mail.isp.com. (priority 10)
MX - backupmail.isp.com. (pririty 20)

www - serveripaddress