Looks like the Moorings would have you starting at Highbourne Cay, at the northern end of the Exumas. This is a swanky resort island, with a megayacht marina, and services to match. You can get groceries and liquor here, but I imagine they are pricey and limited -- and you won't be able to get anything else -- no food, water, ice, fuel, restaurants -- until you travel substantially south (Staniel Cay, or Black Point settlement).
There is a charter outfit outside of Nassau called Navtours that also allows Exuma charters. They market to a French Canadian market, which is why you don't hear much about them. We sailed with them this May, crossing from New Providence to Shroud Cay, in the Exuma Cays Land and Sea Park (aka Paradise) and ending at Staniel Cay (and paying the one-way charter premium). New Providence island has a lot more choices for provisioning, and it's easy to get a flight to Nassau from the U.S. I expect charter rates are more "reasonable" as well. (My trip report is here:
Exumas Charter)
However, in addition to the points Mike K. lists, you need to add the consideration of the crossing here. In June, the weather is not likely to be an issue impacting your crossing, but despite chart plotters, you've pretty much got to have someone on watch while crossing the "Yellow Bank" to watch for uncharted coral heads. If you're stuck motoring (as is likely the case), this is a 7+ hour trip.
The Exumas are the stuff of beach lover and escapist fantasies (swimming pigs and the tourists they attract notwithstanding....). However, for a first time charterer, I would strongly suggest the Abacos over the Exumas. They are less charter-oriented than the BVIs, but definitely boat-oriented, with line of sight navigation, shallow and protected waters, and lots of amenities for sailors. Also, several charter companies from which to choose.