Your travel is going to be huge. Seminyak to Lovina is about 3.5 hours driving. It is a nice drive though. Then Lovina to Gillis is a 4hr drive to the airport, 1/2 hour flight, 2 hour drive to Bengsal (sp) harbor, 20 minute boat trip. Or 4 hour drive to Padang Bai, 2.5 hour boat trip which can sometimes be unpleasant. Amed is 2.5 hours from the airport or 1 hours from Padang Bai. Either way 4-7 hours travel depending on what mode you took. Basically 5 days of your holiday are going to be pack up drive unpack days which is not what I would do

. Plus imho the Gilli's is a very very overrated dive place. Hmm ok it is rubbish compared to Komodo, Lembongan, Southern Lombok, Menjangan Island, Amed, Tulamben, Pandang Bai plus the snorkeling is crap as most of the shallow corals are bleached out. Just saying, some people do like it and wont agree

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It's always a hard call recommending things but here is what I would do for 21 days. Hmm ok well I am in August September

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Long flight get off you just want to unpack and relax so start at Sanur. Nice intro place close to the airport and easy to get to traffic wise, great food, lots of it and spend a couple of days there to unwind.
Head over to Lembongan. I use World Dive. Insane diving not much to do other than dive, snorkeling is really good, some good restaurants. Very relaxing place. The 25 minute boat ride leaves from Sanur so all nice and easy. I use Rocky Fast Boats to get to and from this island.
Then head back to Tulamben. Amed is good Tulamben much better, well easier. Even if you don't dive anywhere else it has enough to keep you going for days and it is all right in 500 meter stretch. Amed is a 10km stretch and the dive sites are spread out. You can always do a day trip to them from Tulamben. I usually spend 5 full days there and that is not enough. It is my favorite place to dive due to the variance and ease of diving.
Next stop Ubud. Here is where you do most of your tours, eat incredible food and start buying stuff. 3 -4 full days there easy. Also lots to see on the way so the trip there is not wasted. Trust me rice fields and mountains get a bit meh after a couple of hours of stop start traffic so I use travelling between places to see other things which saves taking a tour on another day.
Then end the holiday in Seminyak. Close to the Airport, bit upmarket yuppyville but good restaurants and bars, good shopping and only a short Taxi ride into Legian and Kuta for the full on shopping places. We always end in this area so we don't have to lug stuff around for the rest of the holiday.
The above is a circle so no going back on yourself. Bali is small but travel can be painfully slow. Hope this helps a bit.