Aloha JI808!
The shotgun ranges have the most more structured operations in Kokohead. There isn't really any opportunity to "just practice." You go out with a group of up to 4 other shooters, and shoot a round (25 birds). You pay for each round, in the $7 range (depending on if you're talking about skeet or trap, and whether or not you are a member, the price varies a bit). Skeet operates Sat & Sun; trap operates Sun only. And there is a LOT more waiting at the shotgun ranges.
Trap only has one range, max of 5 shooters. So, with 30 people there, the most frequently you can expect to shoot is every 6th round. Skeet has 2 ranges, so the wait time is less, but you're still tied to intervals.
With the 10/22, you can shoot both at the state-run rifle range and at the club-run silhouette range. The silhouette range is only opened Sat & Sun, while the rifle range, as you noted, has a few hours on weekdays as well.
Personally, I'd start with the .22. The shooting is more affordable (cheaper ammo plus no "pay to play"), and there is more range availability. Then, when you do buy your shotgun, take both to the range. Put your name on the wait list at a shotgun range, dash over to the silhouette range for a quick box of ammo, then dash back! Not only will you get more shooting done in a day, it's great cardio!!