If you get a server from a budget dedicated server provider, then it is likely all you'll get is a machine with an O/S of your choice installed. It will be up to you to choose and configure the various services you wish to offer. If, as you say, you intend to get a fully managed server, then unless you plan to hire a third party, your listed choices might be wrong for you. If you do intend to hire a third party, and lease a machine from an unmanaged provider like LayeredTech, then make sure this future admin knows what they're doing. In this case, they'll not just be providing daily support and maintenance, but also possibly choosing which daemons to install (eg. Sendmail or Postfix, etc) to provide the necessary services.
Email can be one of the less enjoyable services to administrate but, depending on the number of end users, it might not be that bad. They should be confident about their understanding of the configuration, since an incorrectly configured mail daemon can lead to abuse, which can have a domino effect on all your services (getting black listed, blocked, etc) if not caught quickly.
Most hosts will offer a control panel (Plesk, Cpanel). It is quite easy to administer email using these control panels - all point and click. If you have problems, your host will walk you through it.
In answer to your questions:
>Will I need an email server?
Yes, if you want to handle your own email. (There are 3rd party services that will handle the email for your domains if you don't want to.) You probably know this, but just to be sure: usually, when people talk about a mail server, they are talking about a software application that serves mail. Unless you are doing something special, you probably don't need another physical server to handle mail. That is, you will probably use the same machine to serve web, mail, etc.
The mail server software will come with just about any machine and will probably already be configured and installed.
>How reliable are email servers?
Compared to what? It usually is not really a concern. The mail server rarely crashes - in fact, I don't think I have ever had it happen (it is more common for a web server or database server to crash).
>Can you give me any resources on the web where this is outlined?
I assume you are referring to the question "how will I handle the email accounts". Just so you can get an idea of what is normally involved: the following URL shows how you would setup an email account using Plesk (it is similar for other control panels).
The control panel handles most of the details, but tt isn't all that hard to setup mail accounts manually if you are so inclined.
>I am not a hosting expert so I will be using fully managed servers.
Note that (as far as I know) LayeredTech does not sell managed servers, although they have resellers that do. I think DedicatedPlace is a LayeredTech reseller and, they do offer managed servers (and they get good reviews).
ThePlanet servers are not technically 'fully managed' although they do have extremely good support and, for example, they would help you setup your mail server.