Just finished my third trip to Bali. We stayed at Grandmas Plus Hotel in Legian, near Kuta. The hotel is pretty basic but very good. For ubud, even i couldn't find myself a good hotel and we finally stayed in a private room in a hostel. Give more days to Ubud, you will mostly like Ubud more than Kuta. Last time me and my friends stayed in Chili Ubud Cottage in Ubud, but that was just okay.
From Kuta: 1 day for in and around Uluwatu (several beaches are there) GWK. Go for Kecak show in Uluwatu. Its good. Visit Uluwatu around sunset and go for Kecak after that. You have free shows of different artforms in GWK every hour as well.
1 day for shopping/ Activities/ roaming around/ Tanah lot, if you want. Visit Tanah lot in the evening.
1 day for Nusa Penida Boat trip. If you want to hike down to the beach, go in the early morning before the rush.
From Ubud: 1 day for Ulun Danu/ Jatiluwih rice fields.
1 day for Waterfalls/ Tegalalang/ Monkey Forest/ Tirta Empul/ Roaming around Ubud.
Visit the ridge walk/ rice field walk near the Ubud centre, if possible.
Vist the shops near the MAS Village in Ubud if you want to see amazing art work they do in stone/ wood.
If you want, you go for a Barong/ Kecak/ other art forms in Ubud as well. There are evening shows almost everyday near the Ubud palace.
If you're traveling back to Airport from Ubud, make sure you have enough time. Sometimes there will be traffic and may take more than 2h for the commute.
If you don't have a trolley or have only backpack, you can hire a scooter from Kuta itself for the whole trip. Check the odo/ tire condition/ mobile holder in the scooter before finalizing the deal.
There are many non-touristy places that you visit. Its totally up to you.
Let me know if you need any help.
Thanks,
Akhil