I have narrowed it down to .243, .270 and 30-06.
Well that narrows it down, from the lightest cartridge most people recommend to the heaviest.
Let me say that between those two rounds any standard cartridge will kill Bambi just fine. Don't worry about cartridge.
I've used the .30-06 for close to 40 years and each deer I've shot would be just as dead if I'd used any other legal round.
Pick the rifle first. There are more good rifles being made than bad ones. The main question is how a rifle fits.
If you put your a rifle to your shoulder and can look straight down the sights when you place your cheek on the stock, then the rifle fits. Remember that a rifle that fits when you're wearing a t-shirt might not when you're wearing a heavy jacket.
My own choice for rifles is the Winchester Model 70.
I can't think of a rifle that I haven't enjoyed shooting. The Savage/Stevens are very good rifles that don't cost much. You could probably get a good deal on a used one.
CZ USA makes some very nice rifles that might be out of your price range.
The Weatherby Vanguard is a rifle that gets recommended here frequently.
Of course Remington makes real good rifles at a variety of price points.
Since this is your first rifle consider getting a used gun. Bolt actions are really simple machines and if treated well they'll last well over a hundred years. So buying a used gun isn't like buying a high milage car.
Well except for the similarity in salesmen.
If you choose one that's rust free, clean, with a smooth action that locks up tight, odds are it's a good gun.
I am also someone that thinks getting a .22 is a good idea. You get better the more you shoot and you can shoot a lot more .22 than centerfire.
Get some lessons and take the hunters safety course.
Find out were you plan to hunt and take the family there in the off season for hikes and picnics. Learn the land and find out where the deer are.