My fiancé and I are looking into doing a Safari near Mount Kilimanjaro. We would love to climb the volcano and then take a safari after. Does anyone have any recommendations for a safari that was really good? What are typical prices for a safari?

Safar in Tanzania has become quite expensive. There's a lot of corruption and everyone is taking a cut of the profit.
But there are still a few good agencies that you can rely on for a good experience and well priced
You can contact 'Tanzania Roadside Expeditions'

Another good option for a Safari, would be the Kruger Price. Very well priced and you can go places were there is no tourist
Very deep in the bush

I know luxury safaris best. That's how I travel personally and that's what I plan for my clients. For a tented camp, I suggest taking a look at Legendary Nyasi (Jul-Oct) or Legendary Songa (Jan-Feb) depending on the time of year you travel. You could also look at Legendary Mila in June. If you truly have no budget, I was at Singita Sasakwa last month. Here is how good Singita is: I've stayed at several Aman resorts in Asia, Southern Ocean Lodge in Australia, Awasi in Chile and more. Singita beats them all IMO and is, IMO, the ultimate multi-country luxury brand! I've taken my wife to every one of their lodges in Africa. Singita and Legendary would pair well with Little Chem Chem which is driving distance from Kilimanjaro. A safari conducted in a quiet area of the Serengeti without massive amounts of tourists is going to be roughly $1,000 to $3,500 per person per night. If you want to go in the peak season (coolest, driest, highest biomass of animals, most expensive), I would go in September after the hordes of kids from the US and Europe have gone home.