If it is your first gun and are looking at all options don't handicap yourself by just picking those three for choices.
You can add .260 Rem , 7mm-08 Rem , 7 x 57 , 6.5 x 55 , 257 Roberts , a bit more obscure but still options.
OR
.308 Win which is chambered in almost every rifle out there and lots of factory ammo to choose from.Heavy or light.
However out of your three I would go .270 then 25-06 then 243.
I like the Weatherby Vanguard in the budget you kind of mention.
Optics , that is up to your eye , go and look through as many as you can.You will know what is best once you look through and compare. 3-9x40 is most common and very versatile.