I wouldn't take an old HDD from an old laptop and put it in a new one: HDDs have a limited life. The older they are the more likely they are to break tomorrow.
And you are likely to void/disable your guarantee/warranty on the new hardware if you take a screwdriver to it. (I can never work out how to get inside laptops anyway)
What I suggest you do is install the latest version of your preferred linux distro on the new laptop's new HDD (alongside win, if you feel you must).
Then sort out the (possible) problems with the new linux install. Meanwhile you have your old laptop to go back to.
Then connect both laptops to your network, and copy your personal files from the old to the new.
But your question was:
I have a question. If i took my harddrive out of an older lenovo T-series laptop and put it into a newer model, what are my next steps? How can i make sure i have appropriate drivers for the new hardware?
And the answer is that you can't be sure, but you can try it and see what happens. Sometimes it is OK, sometimes not.