Sym - Not sure of your age but you sound young. First off, you should never shoot a rifle without knowing what lies beyond the target nor without a proper backstop.Onr of those bullets could have easily deflected off a branch and went God knows where. Maybe hurting or even killing someone.
Secondly - the 30-30 is more than adequate for killing deer. Millions of deer would have to agree. A 30-06 will not kill them any better than the 30-30 with the shot you discribed.
There are literally hundreds of rifles capable of making a killing shot on a whitetail at 250 yds. It all depends on how much you are willing to spend.
Good, reliable, accurate rifles that won't break your (or your dad's) wallet include Savage, Tikka, Marlin and Mossberg. All bolt actions. My preference would lie with the Savage or the Tikka.
Now for chambering selection. Again, I do not know how big you are. But normally the 30-06 is about as much recoil as the average shooter can tolerate. (Notice I said normal and average) Some can handle more, others less. If you are on the smaller side you may wish to consider the .243, 7-08, .260 or, in a longer action, the .270, .280 just to name a few.
I wish you well in your choice of rifle and chambering.