Where do you live, Topic Starter? I've been in fire for a bit and might be able to point you in the right direction.
I've been in wildland fire for 6 years now, 5 years on an IHC (or "Hotshot" crew). I'm now a squad leader on a crew for the Forest Service in Arizona. I've fought fire from Florida to Texas to Alaska, canoed across the border to fight fire in Canada, and worked in just about every Western state.
To answer your question, that depends what job you take, as the Grangeville Jumper said. With a job on an engine, fuels crew, or district crew you MAY have time to train, but not consistently. If you take a job on a Hotshot crew, you will have very little time to train. You will train during your 2-weeks of critical training at the beginning of the season; and you will have a couple days off to do as you please every 14-22 days or so (depending on travel). You will most likely be too tired to run, however.
Let me ask, what do you hope to get out of your experience? How much are you willing to sacrifice? If you want a paycheck and need to continue to train, I would advise that you pursue something else. Fighting fires, especially at a high level, is not for you. However, if you want the greatest physical and mental challenge of your life, then this might be for you. It could very well be the most rewarding and enriching experience of your life. You will get to work with some amazing folks, in amazing places, and you will see and do some incredible things.
I can't encourage you enough to try to get on a Hotshot crew if you do decide to follow this path. There is absolutely no substitute for seasons of experience on a Type-1 crew. They get the most time on assignment, and are given the most physically challenging, specialized assignments. You might be awake for 30-40 hours straight, putting in handline and pulling off a massive burn-out operation to save hundreds or thousands of homes. After many/most shifts, you will be an aching mess. But you will develop very tight bonds with your crewmembers. After a season you will be closer to some on your crew than you are with your lifelong friends. Something about sharing a mission-oriented purpose and living and working in a very stressful environment will do that. Pretty amazing.
It is a job, still, no doubt. And you can make a good amount of money. GAC Jumper's number might be from seasons past. A season with 1,000 hours of Overtime (pretty standard for a Hotshot crew anymore) works out to be about $40,000. Pretty good, yeah? Take into account that you won't be buying food during this time, no need to pay for rent, etc. The Forest Service or BLM takes pretty good care of you. Most of your money you won't have time to spend. At the end of the season, you should have a good chunk saved up. And now you have 6 months off to do as you please. Not a bad deal. Now, you might make less than this, or you might make more. My buddy made 1400 hours of OT during a season on a Hotshot crew. He went to Oklahoma in March and got 200 hours of OT more and grossed $84,000 that year. Or, you could get on a crew in the Northwest or Montana and be getting snow in June and not make squat...It depends.
I don't want to type anymore. Hope that clears somethings up. In short, if running fast is the most important thing to you right now, don't bother applying. If you are interested we can talk further. I can help you in your pursuit of a job, but it may be a little late for this season, but people ARE still hiring, but you need to get your ass moving. Give us some more information about "you." Here is a link you may find helpful:
http://www.fs.fed.us/fire/people/hotshots/IHC_index.html