I can't help but think you'd be plenty well served with a .243, or a .22-250. Hunting varmints and target shooting is quite a bit different than having a light rifle to hump around all day. The .243 is good because it can do well with light 55gr, 58gr, and 70gr varmint bullets- and double duty if needed with 100-105gr hunting bullets.
Bullet selection is, or should be, a major consideration. In areas where you might varmint hunt- ideally you don't want to zing a hunting type bullet through the critter and out-n-over the fields and pastures. A varmint type bullet will be more likely to break up and/or stay inside the critter. On a shooting range, that shouldn't matter. At 500yds and on a range- heck, even a .223 will do that. What you don't want is to form a notion that any particular chambering, cartridge, caliber, etc will make you a great shot. If you don't have a decent rifle, glass, and mounts to start with- then how do you know if you're shooting to your true potential?
If you've limited experience with a rifle and the Rem Varmint is in your range- then by all means go that way, but don't expect to like it very well once you have to walk & stalk with it. Still, I offer the suggestion of .243 chambering. Savage, Remington, and CZ are all great places to look.
On the CZ- read up on those. They have a single-set trigger and come in heavier weights. Very good rifles!