My suggestion is to rent a car. Mostly anything will do, but a Toyota Landcruiser is nice for the dirt roads and comfortable for hours of driving. Next up, drive to Reykjavik. On the way stop by a supermarket to buy supplies and then it's off for breakfast at Bergsson Mathús, Downtown Reykjavik (thats me and maybe too much). Then I get the hell out of town and start driving. Totally doable is the Fosshotel Glacier Lagoon (amazing food and accommodations, whisky on glacier ice, etc.) with limited couple of stops along the way, enough for a first impression and plans to come back. Most large waterfalls are right by the road and traffic shouldn't be bad that time of year.
Like this you're right between Skaftafel (Glacier hike) and Jökurlsarlon (Glacier lake boats) and a drive to Stokksnes takes about 90min and Höfn is a nice/only lunch spot.
On the way back stop at Katla Geo park and Reynisfjara Beach, the "Black beach Cafe" is great
Do some research for outfitters, we only went for ad shoots and had a production company taking care of us.
As the weather is changing constantly, ask at the hotels and gas stations and take it serious, we had storms rolling in under an hour, highways closed for 24hrs, etc. I would plan on being stuck for 24hrs somewhere and stay in Reykjavik at the end of the trip, maybe 2 nights (we did Hotel Borg and 101, both splendid) and if you don't make it to the first you at least make your flight out (3 out of 3 trips we had weather issues).
And bring payers, rain shells top and bottom are nice to take off...