Start competing as an amateur as soon as possible.  After a year of lessons, you should be able to compete and do well in beginner competitive levels.  If that's not the case - maybe reconsider your training routine and habits, possibly look for stronger teachers.

There are 4 major dance styles in the US. International Standard, International Latin, American Smooth and American Rhythm. Chances are, you've been learning smooth and rhythm so far (because you said Arthur Murray studio). Each of them includes 5 dances except smooth.
The dances are
Standard: Waltz, Tango, Vienneze Waltz, Slowfox and Quickstep
Latin: Cha-Cha-Cha, Samba, Paso Doble, Rumba and Jive
Smooth: Waltz, Tango, Foxtrot and Vienneze Waltz
Rhythm: Cha-Cha-Cha, Mambo, Rumba, Bolero and East Coast Swing

In the US, there are no requirement to become a ballroom professional. If you claim to be a pro - you're a pro. You don't want to look ridiculous next to other competitors out there if you chose to compete, so you should choose a level that corresponds to you ability.  

Roughly speaking, there are two kinds of professionals. One kind is social dance teachers. They usually don't compete or never have competed. In this case, ISTD syllabus exams and certifications would be a big plus in terms of possible employment. The other kind of professionals competes or at least used to compete at professional level. If they are actively competing, they didn't touch the ISTD exams because neither they have the time for it nor the need. Their competitive record is good enough of a credential. These people usually practice their professional dancing in the morning and teach in the evening.

Blackpool Dance Festival (UK) is probably the most famous competition in the world.
USA Dance National Championships
Ohio Star Ball
and many more.

Also check http://www.usabda.org for amateur competitions near you.
You don't need major competitions if you're just starting out. Find a local competition, it's a lot cheaper to attend, and a lot easier to dance. I traveled to competitions and competed locally - there is no comparison just how much harder it is to do well after you traveled across the country, spent a night in a hotel, spent your morning trying to locate the competition venue and your partner who will always disappear on you in this case, and in general getting your act together.
Good luck with all that.