I did exactly what Phil suggests in attending the DSC convention so that I could look several outfitters in the eye and get to know them and them to know me. I think you will find that any of the outfitters / PHs on this site would deliver what you want. But, we all have different personalities and finding someone you hit it off with is important as you are going to spend 16 to 18 hours a day with them for your entire hunt.Most new to Africa hunters start with a plains game (PG) hunt. Not all but most. It's a good way to get started. I say that because once you've been, you'll want to go back. But why start with PG is that you don't have quite the layout of money that you do with a dangerous game (DG) hunt. And it gives you a chance to "get" it.
South Africa and Namibia will offer you great PG hunting for reasonable prices. You'll be able to hunt multiple animals for less than many Alaskan hunts which are for just one animal.
As to which animals you want to go for, I'd recomend looking at the websites of the various outfitters you see here on AH. The websites usually have a "pictures" section where you can become familiarized with the various species and see what stirs your stew.
As for DG, there are so many options depending on what you're after and how fat the wallet may be. Yes Tanzania is a great place to go for both DG and PG, but also very expensive. Zimbabwe also offers great DG hunting and for a much more reasonable price. Then there's Mozambique and the list goes on.
I would strongly recommend attending one of the two big conventions, either Dallas Safari Club or Safari Club Int'l. You can visit with the various outfitters who exhibit there and get a good education on what's offered where and for how much.