I am a season passholder to Great Adventure. The park has gotten much cleaner over the past few years. Security guards and even management walk around the park with grabbers to pick up any trash on the ground. Operations have also been better. Maximum trains are running on coasters all the time even when it isn't crowded.
Although a weekday in June probably won't be too bad, getting there at opening is important. Usually from 10:30 to 12:00 the park is empty.
Frontier Adventures opens at 11:00 a.m. and that is where the boarding for the safari ride is. If that ride is important to you, I suggest getting to that ride right at 11:00. It is very popular and low capacity and even on a day like Saturday with bad weather and low crowds, it still had an hour wait.
The rides with the longest waits are Kingda Ka, El Toro, Superman and Green Lantern.
Also don't go by the wait time signs, those are only accurate when 1 train is running. With Nitro for example, when the line is at the 30 minute from here sign, with 3 trains running it is usually just over a 10 minute wait from that point.
And since this is the budget board, I also have to mention prices are horrible there. Parking is $22 for regular, $32 for preferred (I paid $60 for season pass parking). A burger and fries is $9.99, chicken fingers and fries $10.99 etc. I used to never eat in the park and stop at WaWa outside the park. This year I do eat in the park since it was $75 for a season long dining pass but for a 1 day visitor that won't help.
Here is what I typed up on another site when someone asked what order to ride the coasters in but doesn't factor in the safari:
I would ride the coasters in this order: Superman, Green Lantern, Kingda Ka, El Toro, Rolling Thunder, Bizarro, Runaway Mine Train, then take the Skyway over to the Dream Street station, then go to Nitro, Batman, Dark Knight, Skull Mountain and Blackbeard's. Even though Skull Mountain is next to the Skyway station, I put it later since it is more important to get to Nitro and Batman before it as Skull Mountain has a very fast moving line and rarely is more than 10 - 15 minutes.
The alternate option is to go to Kingda Ka, then El Toro and then back to the Boardwalk to ride Superman and Green Lantern but that involves more backtracking.