I would start by getting a really solid scope. I dont know anything about the feild and stream model your mentioning, so i dont know if its good or not, but Im betting its not what you want for varmits at 600 yards.
Something with turrets is best, but a solid mill-dot reticule will work. i bought a Vortex PST for my .243 and am extremly happy with it. If its a stricly long range varmit rig you could look into SWFA fixed power models or bushnells fixed power stuff.
Remember that if your shooting a 1.5" group at 100 yards, if you do everything absolutely perfect, that same group will be 9'' at 600. That being said, we should know your group size on average (not the best you've ever shot). If its not up to snuff you can get the trigger worked and the action bedded.
Handloads are essential for long range shooting. Walmart brand cartiges most likely wont cut it. There are custom'ish ammo manufacturers out there. Try some stuff from Black Hills if you dont want to reload (if they make ammo for the .243 win).
Laser range finder. Most cant do what they say they can, so get something really good. Bushnells work sometimes, but I have 0 faith in them. Leica, Swaro, and Ziess are the go to brands.
The cost of Long range shooting and hunting is emense. But you can do it affordably over time. I would say start off at 300-400 yards, where you start seeing significant drop and wind changes. for those ranges and even to 500 yards, reticules are extremly effective. That way you can probably use the scope you have on it until youve learned some basics. This is where you learn your marksmenship.
I like your choice in caliber. I just bought a .243 win myself (well 6 months ago). Its a plain jane Rem SPS. I bedded the stock, floated the barrel, developed hand loads and turned the stock into a thumbhole style. Topped it with a Vortex PST. took it coyote hunting this morning and saw no dogs so found a couple 1-1.25MOA rocks at 780 yards that needed to turn into gravel.