Welcome to the forum and hopefully Puerto Rico soon.
Most of the island meets those needs, but it all depends on the life style you want to live in PR. Also what sort of company you are starting or bringing with you.
If you need to impress visitors, probably I would add Fajardo, Condado and Isla Verde (by the Airport) and Hato Rey to the list provided by KW, but if you don't then there are a heck of a lot more places.
You can live in the mountains in a semi suburban or country setting, live by the beach or a metro type environment. It just depends what you think will be best for your family and business.
You may want to consider a condo if you do not want to deal with yard work, I say this because you will probably be busy with your company. A lot of condos have swimming pools, some have Gyms also and some may have different types of courts.