Day trips from Kwajalein are severely limited by the access restrictions that govern the atoll. You cannot simply hire a boat and explore; any movement requires authorisation. That said, for those on the base, a couple of options exist.
Ebeye Island — The ferry to Ebeye takes about 20 minutes and runs regularly. Visiting Ebeye gives you an immediate and striking perspective on the human geography of the atoll: the dense Marshallese community that exists in the shadow of the American base, with none of its amenities. It is not a comfortable tourist excursion, but it is an honest one.
Outer lagoon diving — The base dive shop arranges trips to wreck sites around the lagoon. A half-day dive trip to some of the more distant wrecks constitutes a genuine excursion, given the lagoon's scale.
If Majuro is your base (which it should be for most visitors to the Marshall Islands):
Laura Beach — The far western end of Majuro Atoll has one of the better beaches in the Marshall Islands. Accessible by taxi or rental car along the atoll road.
Arno Atoll — The nearest atoll to Majuro, with traditional Marshallese villages and good snorkelling. Day trips can be arranged through guesthouses in Majuro; the journey takes around an hour by speedboat.