If I register through 1 & 1, or any body else for that matter, can I then use the domain name through my ISP such that I can keep it if I move ISP?
That's where the facilities (and prices) offered start to become interesting.
Personally I use a company called Red Fox Hosting (but there are many similar) who provide web space and email services and manage registration of all sorts of domains. I have about 10 domains in total and a couple are registered elsewhere but still have web/email at RFH. That costs about £30 a year plus registration fees.
As a result I'm completely independent of my broadband supplier who I only need (and only use) for connectivity.
I also have a collection of email accounts related to "historical" broadband suppliers (they don't always die when you move on) and use these occasionally.
My advice would be to pick a registration supplier who can also offer at least basic services (e.g. email and/or web) that you expect to use in due course.
Terminology - I tend to call call my hosting service my ISP and my Broadband Supplier is just that.
(my broadband supplier doesn't even provide email or web space as standard)
Oh, and I don't redirect anything, ever. Just access the relevant email servers directly, which of course are included in my hosting package price.