BuffaloBob gave good advise. I can only second it. I own a .25-06 Sendero and for not being a magnum it is very flat shooting. Sometimes it seems to defie the ballistic charts. The .257 WM is only more of a good thing. My rifle likes both the 110 AB and the 115 Berger. Both are great bullets.
I agree with Willys46 on the trigger. Most of the Timneys are do-it-yourself and relatively easy to install. I've installed a couple of them in the past and can say nothing but good about them.
As for scopes: get the absolute best you can afford. I have a VXIII 4.5-14x40 on my 7mmRM 700PSS. I sent it back to Leupold to have a target elevation turret installed years ago and all together I've probably had it for close to 20yrs. On my .25-06 Sendero SF, I have a VXIII 6.5-20x40 that I've only had for less than 10yrs. Neither have missed a bit since I acquired them. I have an older Nikon Monarch UCC 6.5-20x44 (fine crosshairs) on a Winchester 70 HVB in .243 that is amazing. This particular scope is, I believe, in the same class with the VXIII's. Nikon doesn't make this model anymore that I'm aware of and I've had it over 10 years and it's solid.
If you love guns and love to shoot, you have to reload. It's a no brainer. One enables the other. I've been reloading for 25yrs and still love it.
I wouldn't worry about a new barrel just yet. Get the machine marks outta that one first. It's probably got a long enough barrel (26inches ??) as is to wring a lot outta that cartridge. The 257 Weatherby is known for long range connections and lazer like trajectory. Have fun. JohnnyK.