If you are trying to establish a new site, you may not even know just yet what your eventual requirements are going to be. Most new sites start out pretty small and then grow as they attract more visitors who share the address with their friends and the process continues from there. Should you have ambitions to build up a site that will have a lot of traffic and that will be able to use sophisticated web technologies in interesting ways, getting dedicated managed web hosting is a good first step on the road to your goals.
When you get a dedicated web hosting server, you can feel confident that there will be no one else sharing the same machine who could interfere with your ability to keep your site running well. On a shared system, there is always a chance that someone else will be doing some heavy processing or will get a large spike in traffic that pulls a lot of resources away from what you are attempting to do at an inopportune moment. If you absolutely must be sure that your system is always responsive when you need it to be, going with dedicated web server hosting is the obvious choice.
Whether you need Managed dedicated server hosting will depend on how much information technology expertise you already have on staff and whether you want to have people spending some of their work time worrying about managing a server. It is actually effectively cheaper in many cases to pay the hosting company to deal with the server management because they are managing a large number of other machines and can take advantage of the economy of scale in a way that is not possible for a much smaller operation. With this kind of management, you can basically just rely on them to make sure that things are working as intended from the server aspect, so that you only have to focus on your actual site and offerings.
You will have to look harder for cheap dedicated server hosting than you would for shared or unmanaged hosting, but there are good options out there. Take a close look at all the specs for what you are getting for your money, and make sure that they line up with your actual projected needs. Then, you can sign your hosting contract with confidence.