I lived on Palau for 3.5 years having a photo / video shop in conjunction with the Fish n Fins Dive Shop. Depending on what nationality you are depends on the visa issue. If you are U.S citizens you get an automatic 1 year visitors visa on arrival. Most, check with the immigration department for a comprehensive list, nationalities other than U.S are issued with a 30 day visa on arrival which can be renewed twice at the cost of $50 a pop. After that you will need to leave, do some shopping in Guam or hang with the Mantas in Yap, and then re-enter the country.
You would need a car as to not have transport would then take away the chance to truly see the rest of Palau. Away from the hubub of Koror the place is a gem. Waterfalls, Stone Monoliths and other attractions. Personally I would buy an old Jalopy or a couple of grand if I were there for six months. The cost of renting for that period would be astronomical. You should be able to drive on a US license but best to check with DMV there. If not its just a one hour deal to change over to a Palau license. I bought a mechanically sound Jeep over there which I used for the last 2 years for $2800. There are a lot of folks looking for used vehicles so you can generally sell it on pretty quick too at the end.
Accommodation can be had depending on budget. I would go for an apartment rental at the Emelakl Apartments for around $450 a month for a small one bedroom deal. The place is managed by the owners of the Rock Island Bakery on the main Koror Road so best get in touch with them. Just pop into the shop and ask for Pauline. Commonly known as the Howlie Hilton the apartments are also home to a fair number of resident foreign Nationals and, depending on which apartment you get, offer great views out over Nico Bay.
Seeing as you are instructors you may want to also test the water for a job. If you have languages the better. Fish 'n Fins, Neco Marine and Sams Tours would all deserve a visit in that order due to the importance of languages in said order.
Living budget I would suggest if you only treat yourself to one 'Tourist type meal a week, somewhere in the region of $400 a week. But what about diving expenses? A two tanker costs around $120 per person in todays economy over there.
Need any more info? Just holler.
Cheers,
Mark.