Registrar and Host are different companies. Obviously A, MX, CNAME, TXT, SRV should be changed. However, should I move Domain Name Servers to the Host or keep them with the Registrar?
You could really go with either. If you like the registrar's DNS management tools, then you can continue to use them. If you feel the hosting company's offerings are better or that their DNS service is more reliable, you can switch to them.
One thing to consider is that if you ever switch hosting companies, having your DNS managed by your registrar might make things easier. If you're quite happy with your hosting then this is not as big an issue.
Compare things like online tools, reliability and uptime, APIs for scripting and making automated updates, stuff like that. If you ever decide to run your own nameserver, you might want to see if either company offers slave DNS service, too.
There's not really a "right" answer, so you just want to compare the options which are important to you.