Welcome to the club! The .375 H&H is very addictive.
I can only bow to others when it comes to factory ammo, because I have always handloaded my .375. When I talk to writers who have extensive experience with all kinds of African game, the Hornady DGX seems to be the most frequent choice--which also coincides with my research on brand/load popularity (I work in the industry).
My personal recommendation is to use this excellent purchase as a bridge to handloading. Besides a modest initial investment on press, accessories and components, it is a very rewarding activity that opens up a virtually limitless number of possibilities for you. (It would also be your official "Powderhead Club" card!)
If dangerous game is not immediately on the menu and you want to turn your .375 into an almost all-round rifle, I can tell you what's worked for me: a 270gr spitzer traveling at about 2500 fps: pleasant to shoot, accurate, flat enough trajectory out to 250-300 yards, no ruined meat and good penetration, and very little in excess of what you need from whitetail to anything in North America and most of Africa.
With the 300gr, you start to have a gap between the excess in recoil and what you really need for animals below the Cape Buffalo (or perhaps,
perhaps the biggest Alaskan bears). Your mileage may vary, of course. I've never tried the 250gr, so I will defer to others with more experience with it.
As far as cleaning the rifle, I use the tried and true method: jags, brushes, cotton patches, Hoppe's N. 9, and oil. Recently, I've replaced the last two items with what I consider to be the world's best do-it-all liquid product--the 120-year-old veteran Ballistol. While on a hunting trip, I keep a bore snake that I run through the barrel upon my return to camp. As far as I know, they don't make one specifically for .375, but the .35 caliber works perfectly.
Lastly, if you're a member of the .375 H&H club, please know that, traditionally, it's classified as a
medium, not large bore. There is a whole magic, fascinating world of truly large-bore cartridges that one day you may want to explore.... if you dare....