I just got hired as an RN for a local government two weeks ago. I learned through my trainer and other staff members, that the company is currently looking for a supervisor for the shift I am currently working. My trainer encouraged me to ask them for the position, since I have supervisor experience, the degree, and a few ideas that can potentially help the company save money and time; however, I just got hired and I am still in training. Will it be a good idea to ask them to consider me for the position? If so, what should I write on my consideration letter?
I think you've already gotten the answer you need from your trainer. The worst thing that will happen is that they will turn you down.
You've seem to have a good start in the letter for consideration - relevant experience, degree and ideas.
Will it be a good idea to ask them to consider me for the position? If so, what should I write on my consideration letter?
If it's a position you want, then it certainly makes sense to ask for it.
In your consideration letter indicate the reasons why you want the position, and the reasons why you would be a good fit.