There is much to learn and this is the place to do it. Check out the Glossary Under the Help section (look up top) as it will help you understand a lot. There are a lot of choices to make, which airlines(alliance), route, class and more for each flight. booking 6-9 months out should yield you a good priced ticket and help you plan accordingly. Sadly a StarAlliance RTW booking which would give you the most flexible experience and most airlines is out of your price range.

Good advice above!

Also, keep your seat belt snuggly fastened at all times when you are seated, even if the Captain turns off the seatbelt sign.

I often get up to use the lav, stretch my legs, etc., but I always go sit back down as soon as I am done and then immediately refasten my seat belt. I keep it snug. I've had two times (both on transpacific flights) where the flight went from completely smooth to moderate turbulence within one second.

Have a great trip!

JK-SFO

If legroom is a consideration, UA offers Economy Plus, which has a few inches more legroom for a reasonable fee. Also, you can check seatguru.com when you are ready to book your flights to see what the seats are like on the aircraft you'll be flying. I am partial to the international terminal at SFO. On the day of travel, if you find you have a lot of time between flights, you can purchase a one-time entrance into an airline lounge to get a more comfortable waiting experience including snacks and beverages.

Can't emphasize this enough. Such a small thing to do to help keep you safer, yet so many people don't. I've yelled at relatives on flights for doing it and made them refasten their belts, getting me a kudos from the FA. Amusingly enough, I've heard a few other "clicks" afterwards too, as others re-fastened their belts after hearing my comments.

Looks like some good advice from the above posters on routes and other considerations. With a $2500 budget you should be able to get a nice ticket for yourself.

Above all remember that it's supposed to be fun! Travel's not something that should stress you out.

I didn't read everything totally so forgive me if this was covered. You asked about checking luggage. If all of your flights are on the same airline or alliance (like a family of airlines), you should be able to check it in PIT and not worry about it until you reach your destination.

Enjoy your journey!

Any advice on where to keep a wallet? I was thinking about a coat but in the summer I wont be wearing one and if did have one i'd be worried someone would just steal the coat.

Keep enough cash in your pocket to supply your needs for the rest of the day. If you lose this, it's not a disaster. You may also want to carry a debit card with a limited amount that can be withdrawn before it can be cancelled (or needs topped up).

Keep anything else in a money pouch worn inside your clothes, or even better leave it behind.

Edit: I should point out that I don't personally bother with this, and carry these in my pockets. This advice is only if you're worried or visiting somewhere particularly unsafe.

I think some of these statements are mildly, if not beyond, paranoid. I've traveled quite a bit and in some sketchy/high theft & crime areas and never had a pickpocket succeed. Most have been so bad at it it's not even funny. To carry limited items with you is smart, lock up your credit cards and important papers but carry some cash and a single card to carry. A money belt on your body can be way more hassle than the protection it provides. Also some thieves may see you as a larger target for direct confrontation vs pickpocketing. The seat belts while sitting is common sense on a plane but if someone, even a family member, turned to me and lectured me about it I'd think they were suffering from oxygen deprivation. I've been on planes that did massive drops through pockets in the airstream and never been injured. Am I that lucky? I doubt it. Air travel is extremely safe and mostly you just have to use common sense and relax. The more calm you are the calmer people around you will be.

I should have made it clear in my last post that I feel the same way, and carry all my cash, several credit cards, driver's license, and pilot's license in my wallet which is in my front trouser pocket. My mobile phone is in my other front pocket, plus (in countries which require it) my passport. I've never had anything stolen, and by far, overwhelming, enormously, the biggest risk of me losing valuables comes from me setting them on the table or floor in somewhere like a restaurant and forgetting to pick them up again before I leave (which I repeated do with my hat).

My advice was purely for those who are feeling worried or who are visiting places with a particular reputation for pick-pocketing (Naples, perhaps?).

Drink lots of water in-flight or get a headache. Bring some Tylenol PM for long transpac or trans atl flights.