What would be a reasonably priced (cheap im poor lol) tent, sleeping mat, etc., that will deal well wih Hawaii's climate?

namaste dude. greatly informative question by stating where you are in the world, Kudos to you.
And good luck & mana to you and yours for the journey.
to avoid my good wishes getting deleted I better attempt an answer…. try searching online for your local camping stores and compare prices.

There are a lot of recommendations for Bonanza, which is in Oakland.

A lot depends if you will need a tent to backpack and be mobile or one you set, forget and use like a base camp/place to stay. Check Goodwill for used gear, Craigs list but what ever you get the rain fly will be every thing as well as a vestibule area for storing wet gear.

Either way first go to a reputable sportings good store and have a sales person teach you the basics on good gear for your needs.

Try reioutlet.com

This time of year you could wait and buy a couple of blue tarps, some rope and an el-cheapo sleeping bag to rig a hammock and fly. Total about $25. If you are more than one, instead of a hammock, buy the tarps, use one as a groundcloth and another sleeping bag. $40. I’d buy them at Wal Mart once you get there.

http://www.campmor.com always has clearance going on.