Assuming you stayed in the US all year and meet the substantial presence test, you are all resident aliens. Turbotax does not list resident alien as an option, so you would choose legal resident, which is the same for tax purposes. In order to claim the children as dependents, you would need to apply for ITIN numbers (the same for you, if you are not eligible for an SSN). To apply for ITINs, you need to fill out form W7 for each person needing one and attach the required documentation:
https://www.irs.gov/pub/irs-pdf/fw7.pdf
Instructions:
https://www.irs.gov/pub/irs-pdf/iw7.pdf
You would submit the W7 forms together with the tax return. Make sure to file on time (or even a month before the deadline), as you would not be eligible for the child tax credit without an ITIN at the time of the filing deadline (or the deadline for an extension, if you apply for an extension).
In order to use Turbotax to prepare the return itself, enter "dummy" number starting with a 9 for anyone without an ITIN/SSN. Print your return and white those numbers out and write "Applied for, see attached W7" over it.
