It really gets to me when someone calls Bali 'bogan'. Being Balinese, it's quite confronting how tourists treat our island.
These suggestions such as 'getting away to Canggu' (which, incidentally, is actually in the area of Kuta North, therefore not out of Kuta), or The Nusa Dua, don't really solve the problem. The Nusa Dua is actually out-of-bounds to anyone not staying there, meaning they have restricted locals from entering, and even enjoying the beaches there. It's awful how they were allowed to do this.
There are hundreds of villages, away from the tacky tourist hotspots of Kuta, Sanur, Nusa Dua, Jimbaran, Uluwatu, Ubud and Canggu, where we love having tourists, and you will receive a
genuinely authentic experience. You can still get transport to all the main tourist hotspots if you need it.
My suggestion would be to go near Ubud not in it), and be open to experiencing accommodation that isn't five-star resorts (which leads to us, in the smallest villages, running out of water and electricity in peak periods because it's been drawn by the major massive hotels and chains).
And don't fall into the tourist trap of paying for things like yoga and traditional events that are actually organised by foreigners! In a Balinese village you will experience them, and much more, free. It is quite wrong, in fact, to pay for them, and all too often expats and their companies try to make a profit off unsuspecting tourists who pay exorbitant prices for a 'traditional' experience.
Learn a bit of Bahasa Indonesia (the national language) and Balinese (the native language) - which are both distinct languages and quite different from each other. People will love you for it. But most of all, be open, friendly, and enjoy our island, which is absolutely not bogan!