If I register my business name/logo/slogan as a trademark, does that mean no one else can use it?

Not necessarily.  Trademark law is intended to protect businesses from unfair competition and consumers from confusion about the source of goods or services.  Trademark rights only protect your mark in relation to the specific goods or services that you provide.  The USPTO often registers seemingly similar marks when the goods or services are different and there is no likelihood that consumer would be confused as to the source of the goods or services provided by the marks.

For your business, before you consider a trademark infringement claim, consider that not only does the potentially infringing trademark have to be similar in sight, sound, appearance, and connotation, but the goods and services must be similar as well.