Mail and Hosting problem

[wesen]

Your name servers just need to point where your website is hosted.
For mail you need to configure the MX records and associated SPF and DKim records where applicable.
Your mail provider will give you these records.
Technically you can actually have a seperate provider for DNS services as well but this is really aimed at people with nothing else better to do with their lives, or larger enterprise clients where downtime isn't an option. For most regular cases having your domain name your web hosting and email hosting with different providers is fine, heck most of the time you can get away with one provider for both hosting and email if you really want, depending on the setup.