Excellent question!
To travel around the world in 6 months is a short amount of time but it's not impossible if you fly a lot and go to countries with good transport infrastructure. Since you are from Fort Lauderdale, Florida, I presume you are looking to places different from home (less beaches, coastal resorts and more 'inland' scenery, cultural sightings and maybe snow?
Month 1: I'd fly to Santiago, Chile (1 week). Cross to Argentina in direction of Buenos Aires (2 weeks with a long weekend in Montevideo). Take time to visit the Andes (1 week). It will be late Fall, try to ski. You don't have big mountains like that near Florida! Fly from B.A. to South Africa. There are direct flights to Johannesburg.
Month 2: South Africa for 3 weeks. Then Egypt for a week (there are direct flights from Johannesburg to Cairo almost daily). Why South Africa? It is easy for first time traveller to Africa. It has good infrastructures. The National Parks are of quality if you want to discover the african fauna. It's not cheap but you will be going in the winter season. It won't be hot. Egypt, in a week, you can cover the pyramids, Cairo and a ride in the desert or a trip to Alexandria but I wouldn't try to squeeze the valley of the kings or the red sea on that tight schedule.
Month 3-4 Fly from Cairo to Istanbul. Fly Istanbul to Rome and go north from there to St Petersburg. My itinerary would be something like: Rome, Milan, the Alps (italian or Austrian), Vienna, Bratislava, Prague, Warsaw. Take a low cost flight to St Petersburg.
Month 5: Transsmongolian train to Beijing from Moscow. I wouldn't stop in russian cities and go straight to Mongolia where i'd try to spend a week or so off the train. Enjoy the longest train trip of your life and make friends on the train.
Month 6: China (3 weeks) and Tokyo, Japan (last week of your trip). Just pick 2-3 destinations in China or else, it will feel like a marathon. In Tokyo, on your last night, book the best hotel you can and threat yourself because you'll spend a day on the plane the next day. :P
With USD$25.000, you shouldn't have much problems. I'd skip India and South East Asia for that trip. Why? Because those are destinations you must go slow. If you go fast there, you will hate yourself.
As for Europe, you can skip big western european cities likes London and Paris. You will have the opportunity to go again. Those are good destinations for later in life when you only have a 2 weeks vacation.
In Europe, try to couchsurf whenever you can. South America, Russia, China, sleep in hostels (book them in advance for Russia and China, it will help with visas). Your big splash will be South Africa. Book yourself a couple nights in a safari lodge. You won't regret it.

