Platform
Obviously, your choice of platform will depend to a large extent on the types of applications you are using and the skills and knowledge you already possess. The two most well known operating systems are Windows NT and Unix (which includes the Linux, and Solaris platforms). Windows NT, the more expensive option is regarded as the most user friendly and easiest to install, especially for those who use Windows on their PCs. Unix is cheaper, but there is usually a much steeper learning curve for those who are not familiar with the more arcane Unix environment.

Data Transfer
Most dedicated server providers will allow you to choose your level of data transfer, usually in gigabytes per month. Usually, you will be paying for this, so you do not want to purchase more data transfer than is realistically needed. This can always be increased as needed.

Data Backup
If you run a site, which is constantly being updated, you will need to back it up frequently. This can be a hassle. Many dedicated hosting providers will provide a back up service for you - usually for an added fee, but the convenience may be worth it.

Monitoring
Your server will need to be monitored constantly to prevent service interruptions. Check to see that your host can provide such monitoring, and how frequently it is done (eg every 5 mins), and what measures they use to deal with problems, which are detected.

Automation
As mentioned, running a dedicated server does usually require a greater level of technical knowledge than shared hosting. However, those who lack technical expertise may still be able to operate a dedicated server --- if the host offers some form of web-based automation to simplify the process of managing a server. Check to see if your host can offer such automation (if you think you may need it).