I have taught many would be firefighters who took a paramedic class.
I don't think it will be quite the change you hope for. So let me get out my crystal ball and look into how this all plays out.
You take your EMT class. It is easy and you pass with flying colours. You're in for $1500. at minimum wage (what you can expect in the current clime for ~120 hours of class and everyone who thought it would be a quick career change) You now have to work for 200 hours to make up that money. Half what you are making now.
What will you be doing? Your old job. so you only have to work 100 hours.
Otherwise you will enter the often unknown world of Interfacility transport. You will be the low guy on the totem pole, lifting people up, putting them on the cot, driving them somewhere, and putting them back. You will not always get a break, you will be rushed in the goal of always getting more transports, not given the tools to do a good job. Depending on your area, you may never see an emergency ever. Ask around, life will truly suck while you hope somebody is hiring firemen and getting into the testing and turned down routine of that. (sometimes it takes years, many never succeed)
Now you will definately see the need to be a paramedic. Not only to help you on your fire test. But you can not compare paramedic class to EMT class or any other class you may have had. It takes total commitment. You will be responsible for material not covered in class, as well as background things such as chemistry concepts you may never have had. It is my opinion anyone who can complete paramedic class could complete medical school. The intensity doesn't change, only the length is different.
After an average cost of between 5K and 12K depending on the school, you now get a pay raise in IFT to let's say $15/hour, more likely around 13. You will be very happy because you just doubled your money. But at that cost. It is 333 to 666 hours to break even of your IFT job.
Now if you haven't got on an FD yet, I will bet money you will absolutely hate with passion going to work everyday. You will start searching out better paramedic jobs. The competition is even more tight for that. ED techs find various level of hapiness.(Usually if you can accept always being the low guy in the hospital) In fairness, my time in the ED as a medic was the best time as a medic I ever had. Flight medic, critical care ground, an telemetry monitor where you stare at EKG screens all day and call somebody if anything changes. You might get the best of luck and score an overseas job. An inudstrial clinic, or if you like ships, something there. Likely you will have to relocate and work for a reputable emergency EMS service. (Which comes not only with a move, but high competition from people across the country)
But don't count on getting one of those jobs. They are looking for career minded paramedics. Who live and breath prehospital medicine. They know a fire flunky when they see them. Having that label creates a lot of difficulty. Plus if fire was your goal, you may never have the motivation and passion you need to be a successful medic. It will be blantantly obvious to all. Who wants to hire a disgruntled employee who doesn't enjoy what they do?
You are back to the IFT game. For the rest of your EMS career. Have fun.
Now don't get me wrong, there is lots to be learned in IFT, and some find it enjoyable. But there is a caviat. The people who enjoy it, work for reputable companies, I assure you in IFT they are very very rare. Also combination services that do both Emergency and IFT have enough diversity to break up the monotony of IFT. But again, good companies are hard to find and hard to get onto.
But your biggest hurdle will be paramedic class. I have seen many "I'm just doing this for a fire job over the years." They struggle, everyday is the test or quiz that can fail them out. They cannot get around the idea that being a paramedic is about knowledge. From their EMT experience all they see is the skills end. (because you can't see a person's thought process) They often do fail and find themselves paying to come back. Those people are truly unhappy. Especially when they fail near the end. Plus they are still chasing the fire dream. 10's of thousands of dollars wasted. Year of their life. Strains on their relations.
Depending on their area they will get a part time fire job or a volunteer job. So they will still have to have a full time employer to eat.
I hate to be the voice of doom. But people just don't understand what they are in for. They certainly don't understand the level of competition for fire jobs and the fallout untill you win that lottery.
In my home area the 4 paramedic programs graduate a total of about 300 paramedics a semester. The local departments hire 1 or 2 people every 2-4 years. Many are laying staff off, and they will be recalled before new ones are hired.
You could get a nursing degree at our local state college for about 24-30K total, they make much better money than the IFT medics. You will find EMS as the "back up" is a very dismal opportunity.
Good luck.