When starting any business or company in the US, I hear you have to register for a business license and do some extra tax forms. Is there anything else you must have in order to run a hosting or design company?

The requirements vary by state. As a bare minimum you'll want to file as a DBA with your county or state (depending on whats required). You should really contact an accountant or business attorney to see what rules apply and make sure you've got your legal bases covered. You can also make use of many companies that provide legal business registration services. Doing a search here should turn up a few of them.

Good luck.