What are the DO and DO NOT's in this situation?
What I will be doing now?
No. What you will be doing now is not important to the relationship between the company (you are leaving) and the clients of said company.
I want to show I am leaving to do something completely different, so my leaving does not reflect poorly on the company.
I'm not sure you need to (I wouldn't unless asked by the clients - and even then, I would confirm with the company, before giving any details).
Where I am going?
I enjoyed working with them?
I enjoyed working with my current employer?
No, it is none of their business.
My personal contact details?
Only if it is acceptable for them to call you. If you had a good relationship with them, they may be tempted to call one week (or month) after you left, and go "remember that module of the gizmo you were working on? How do we fix it?" This may eat into your time on (or looking for) a new job and it may damage the relationship between the company you left and the clients.
You should only provide personal contact if you leave it to friends (not for the client's official records) or if you are mandated in your contract to provide assistance after you leave (and that should always be limited in period).
Send a mail?
You probably should (send an email), but send it to your manager first, and confirm the content with him.
Simply put an 'out-of-office'?
No (you are not OOO).
Call them personally?
That is up to you (and what you decide with your manager).
Your message should be informative, pass on your contact to somebody else and make it clear that your contact information will no longer be valid.
I will be leaving ACME starting next month (January 2034).
Starting from XX January 2034, maintenance of the quibbles will
be taken over by my manager, John Doe.
For any issues regarding wobbles in the quibbles, contact Jane at +0099-99-99-99-9.
The first phrase (about leaving the company) may be missing (ask your manager).