there are many different types of traders on the floor...independent brokers, brokers for large companies (institutions/banks), traders for large/small companies, independent traders trading for their own account...
If you know someone its a big asset...smaller firms like to trust people they know with their money. Larger firms look for a balanced resume, often right out of college.
Many start as runners making dirt cheap wages to then become a clerk and then a trader. Some get in the door through college internships.
I got in from nepotism as a clerk and was a trader the next year. I was 21, w/o hs diploma as well.
Getting a job with a small firm can offer the quickest advancement imo.