Congrats on your new gun! It is adequate for about anything you want to hunt the World over. The only thing you might need more power for is elephant and possibly Cape Buffalo, (and I suppose Rhino if you ever went there). But you can legally shoot them all with this gun and put them down with properly placed shots.
I have that same gun with a VX6 2-12 illuminated that I've used on PG from Duiker to Eland and many in between. And a M70 in 416 Rem Mag currently wearing Trijicon 1-4.... I also own a Swarovski Z6i in 2-12 but if money is an issue, I recommend the VX6. A VX6 in 1-6 would be a fine choice as well.
With the right loads that 375 H&H can be effective to some pretty long ranges so a 2-12 might be a consideration. 2x would be as high as I would go for a low setting on a DG rifle and if your primary purpose is for DG, consider the 1-6.
For about half the cost, a VXR in 2-7 would be a great all around scope and the 1-4 would be a great DG scope, for the money.
Your going to grow very fond of that gun and cartridge. If you are not a handloader (I'm not), you might want to start watching Midway USA, Cheaper than dirt, and possibly get your mouse over to CorBon (10 round boxes) to get deals on ammo. I've found some particularly good deals on closeouts on Midway USA for premium stuff and there was a deal on the Colorado Buck stuff some time back, think that was through Cheaper than dirt.