I would recommend that you spend 1 night in Senggigi, and take a boat to the Gilis from either there, or from Bangsal (which is around a 20 minute cab ride from Sengiggi, and the closest point on Lombok to the Gilis - but be aware, lots of the infrastructure around Bangsal is oriented to running 2-bit scams on tourists heading to the Gilis.

There are lots of hotels in Senggigi, ranging up to a good Sheraton that is, accurately in the ~4.5 star range (although some sites call it a 5*) but it's quite reasonably priced (it typically can be booked for well under $100 a night). Several more economical choices are also available in town as well.

If you don't mind spending an extra couple hours on a small boat in beautiful coastal waters, you can certainly get boats from Senggigi to the Gilis (I did when I stayed at the Sheraton), and that way you can skip dealing with the scammers in Bangsal.

And BTW, I've never been to the new airport (I have flown out of the older Selaparang airport) it's on the opposite side of Lombok from the Gilis - Senggigi would be about 2/3 of the way to the Gilis from the location of the new airport.