KY FF EMT, good questions!
In amateur radio, yes some of the long distance communications are done via repeaters (especially on VHF and UHF.) If a repeater is positioned correctly, it's range can be anywhere between 30 - 150 miles.
Now for worldwide communication there's three routes you can take.
1) HF radio - The HF bands (3 - 30 MHz) have propagation characteristics that allow a signal to bounce off of the ionosphere and travel around the world. The radios that you've been using for EMT are either VHF or UHF, and signals at those frequencies will not bounce off the ionosphere, but instead travel in a straight line into space.
2) Satellites - There are a few amateur radio satellites in orbit that allow for very long distance communication, but have small windows of use as the satellite passes over a given area. Amateur satellites are not in geosynchronous orbit, so you have to keep up with when they pass over head.
3) Internet over radio (IRLP and Echolink) - These are two forms of communication that allow radios to be connected over the internet. This allows your radio signal to go from a local radio, through the internet, and over another radio somewhere else in the world.
Could I have one of my ht1250 programmed to be used?
Quite possibly, yes. One of the Motorola gurus will have to assist you there, but most commercial radios can be programmed to be used in the VHF and UHF amateur bands. Of course in order to transmit in the amateur bands you'll need a license, which brings us to your next question:
Where would one start to get into this hobby?
There are 3 levels of amateur licenses, Technician, General, and Extra. You start at Technician and work your way up to Extra. Each license class give you more available frequencies to use (mostly on the HF bands.)
You take an exam to earn your license. The exam goes through the basics of the FCC rules for Amateur Radio, basic understanding of electronics and radio technology, and safety. There are several books available to help you study, one of my favorite is the Gordon West
study manuals. Once you've studied and are ready to take the test, you'll need to find a local VE (Volunteer Exam) team that will administer the test. Most VE's charge around $15 to take the exam in order to cover the cost of exam materials. This thread
can give you some direction on how to find your local VE team.
To learn more about Amateur Radio in general and understand what we do, check out the ARRL site What is Amateur Radio?
I hope this was helpful. If you have any questions please feel free to ask!