I have a jacuzzi in the home I just bought that worked fine until recently. The jet button (air style) turns on the pump for the jets, but only stays on as long as the button is depressed. As soon as you let go of the button, the pump turns off. What should I look for first?

Likely to be the latching airswitch at the other end that has failed, but could also be the airbutton diaphram leaking or pinholes in the airtube from rodent damage, sun damage or old age.
if you jab the button really quickly and it stays on, this might give you a slightly longer life before the inevitable of replacing one or all of parts mentioned above