They are both good calibers however the .223 is better suited for varmint than the .243. It has cheaper ammo and less recoil. I would go with the .223 for varmints. You can get Hornady factory .223 ammo that is about 3800 fps with a 40gr Vmax. As far as a good lower priced gun I would say go with a Ruger M77 MK II. I have 3 one is a stainless 22LR, one is a stainless 300 WSM, and the last one is a blued .223. The .223 and 300 WSM will shoot 1/2" groups at 100 yards. The only work done to them is the trigger. I put a Timney trigger on the .223 and I did a trigger job to the factory trigger on the 300. Ruger also includes scope rings and it has intrical bases that can't come loose. You can get one for between $460 and $530. They are great guns for the money.
A good cheap scope is the Bushnell Elite 3200. They are about $200 and they have the multi coatings and rain guard which let you fog them up and still see to shoot.
Here is a picture of some of my groups, the top right I shot with my .223 it is a 3 shot group at 100 yards. The third shot is through the second shot hole on the bottom. My brother shot the group under it with the same rifle. His 77 .223 will also shoot 1/2" groups at 100 yards.