If you are experiencing inflammation, the heat will exacerbate the problem. Ice is the answer to help keep down any swelling, which will relieve pressure and therefor pain. If you put your hand on your back and it feels warm, that's a good indication that it is inflamed. Ice alone will not make it better, however. I know the last thing you want to hear after starting an exercise routine with a trainer is that you probably need to rest your back but if this happens every day, you should take it easy until you are better.
I am concerned that the trainer who suggested you put heat on it is trained on exercises but not necessarily in physiology or injury. They may not know that you need to heal. There were already some great ideas passed around. I would hydrate more than usual, especially with a young child and an exercise regimen. I would also recommend that you see a doctor. I know that with no insurance that is a challenge but back injuries are serious. Once you have them, you have them and it's difficult to "cure" for good. You could be more susceptible in the future and the longer it goes, the harder it is to keep it at bay. This is not meant to scare you but to encourage you to seek medical attention to determine if you have muscular or skeletal damage.
If you want a cheaper first step, you may just need an insert in your shoes. My feet are unrecognizable from before I had my kids. There are tremendous changes that take place all over your body. Shoes that once provided the support you required may not fit you the way they used to. I know that some Pharmacies and the larger Wal-Marts have a machine in their prescription areas that you stand on without shoes. It has you stand still, raise one leg then the other, and lean forward, back, etc and then tells you which Dr. Scholl's insert would help you the most. I would start there and try to wear tennis shoes with the insert as much as possible so you can get the greatest result. If that doesn't provide relief within a few days, you should find a doctor.
I hope some of this helps! Good luck!