+1 DWD's link to Axe Murderer. I live and dive in Saipan and I am confident you will be happy with Harry. If Harry is not available Mr. Lee at Dive Saipan is great too.
One of the best dives here is the Grotto. It is an overhead dive but not a technical cave dive. Once you swim out of the Grotto, there are several caves to explore, but again, not technical cave diving. If you're here next month (June), Wing Beach is usually accessible from shore. But if not, it is usually always accessible as a boat dive.
LaoLao Beach one of the most dived sites and can be dived in almost any weather. Also, Obyan Beach is a very nice swallow reef dive (around 50 feet)
Depending how many dives you plan, I'd think about talking a boat dive to Tinian. Tinian is the island just south of Saipan. It is about 45 minutes and a normal dive plan there is the Tinian Grotto (nice but not as nice as the Saipan Grotto) and then the Tinian Dump. Doesn't sound like a nice dive but it is where the US Military push all the B-29 parts and various other WWII stuff into the sea. I am a history buff so I love that dive.
Another cool dive from a boat is Naftan Point. I love that dive. It is a very nice wall dive and is absolutely worth the dive boat cost.
Anyway, that should get you started.