6 day itinerary Austria May 26 - June 1

Sounds like there is no point in having a car as you will be content with staying in the cities. Have you booked your accommodation yet? I'd recommend spending 3 full days in Vienna and taking the train on your 4th day to Salzburg so that you gave 2 and a half days to sitesee there. A decent guidebook will give you all the rest you need.

Which styles and periods of architecture interest you ? Vienna possesses a panoply of them .

While visiting the cathedral in Salzburg, take the stairs(left of altar) down to get a view of the old foundations.

If you are interested in architecture the current exhibition about Otto Wagner, Vienna's most renowned architect, is a must. His city planning and his buildings have shaped Vienna until today.
https://www.wienmuseum.at/en/exhibitions/detail/otto-wagner.html

You have six full days.

May 26 (Saturday) - arrive in Vienna, sleep Vienna; be aware that groceries will be closed on Sunday, so stock up today.
May 27 (Sunday) - explore Vienna; sleep Vienna
May 28 (Monday) - explore Vienna; sleep Vienna
May 29 (Tuesday) - day trip via train to Wachau Valley or Bratislava, sleep in Vienna
May 30 (Wednesday) - depart Vienna and travel by train to Salzburg via Westbahn train (buy tickets on the train); this will be a very heavy travel day as it it the beginning of a four day holiday weekend so reserve a seat.
May 31 (Thursday) - This is a public holiday in Austria; groceries and shops will be closed; explore Salzburg, sleep Salzburg
June 1 (Friday) - explore Salzburg (or Berchtesgaden), sleep Salzburg
June 2 (Saturday) - depart Salzburg

For an architecture-based itinerary, you need to do some research online and determine what interests you.

If you have to reserve seats on the Vienna to Salzburg train, you might not be able to do this (could ask when purchasing tickets), but otherwise get off at Melk and take a look thru the Melk Abbey. Google image it to see why.
In Salzburg, here are some things to consider:

Tour to Hitler's Eagles Nest in Berchtesgaden

Boat trip on Konigsee, also near Berchtesgaden

Tour of ice cave, "Eisriesenwelt" near Werfen

Google image all of the above to see if they appeal to you.

... but otherwise get off at Melk ...

The express trains between Vienna and Salzburg do not stop at Melk.

If you want to visit Melk then you can either take a direct REX (regional express) train from Vienna, or you go with an express train towards Salzburg, leave the train in St. Pölten and continue your journey with a REX to Melk.

For continuing your trip to Salzburg the best choice depends on the time of your departure from Melk. You take a REX either to St. Pölten or to Amstetten, and go ahead from there with an express train to Salzburg.

What type of clothing should you pack?
Weather?

I visited both Vienna and Salzburg without a car and didn't miss it. In Vienna, I think I only used the metro to get from the airport, get to Schonbrunn and get to the train station to go to Salzburg, we mostly walked. In Salzburg, we walked almost everywhere. We did take a Sound of Music Panoramic tour to get to those places outside of town. I used the RS guide and followed most of his suggestions for sightseeing. I'd like to go back to Salzburg in the summer. I was there in January and there was lots of snow. If it is not possible to get to Hellbrunn on public transportation I would probably rent a car for a day. I would say absolutely do not miss the baroque Salzburg Cathedral. I've seen A LOT of churches in Europe. This one is STUNNING! I also really liked walking through St. Peter's cemetery at the base of the funicular. It is beautiful and there is the Sound of Music connection. We did take the funicular up to the fort. The views were wonderful. We walked a lot along both sides of the very scenic river.
In Vienna, we did the Hofburg and did a lot of walking and went into all the churches. Some of the buildings are just phenomenal, look up! One thing I happened upon on my first trip to Vienna was the market (Naschmarkt). Not only was the market fun to wander, but there are some amazing art nouveau painted buildings that front the market. The market is just outside the ringstrasse by the opera. One night we walked along the ringstrasse by the rathuis and parlament. They are lit and were quite pretty. I have not spent much time at the art museums in Vienna. I will say that the Albertina was a good visit, they had a special exhibit of Seurat. Because you are interested in architecture, you might be interested that the Albertina was once a home for Albert and Christina, one of the Habsburgs. You can view some of the state rooms. With your interest in architecture you might enjoy going into the National Library. Many consider it one of the worlds' most beautiful libraries. If you'd like to get into the opera house, consider standing in the line for standing room only tickets. We ended up only staying for the first half of the opera, but we really enjoyed seeing the inside of the opera house and I think it was less than 10 Euro for three of us. If you like wine, you might want to visit a Heuriger. I did that on my first trip and it was very easy with public transportation to get to an area just on the outskirts of Vienna where there are several Heuriger you can visit. There are tours to the Heurigers, you can do the same just by taking a trolley. Have fun!

A small correction to the post above, only to avoid future confusion. The Ubahn (subway/metro) in Vienna does not go to the airport.

If it is not possible to get to Hellbrunn on public transportation ..

Why it should not be possible?
Watch out for bus line 25, which goes back and forth between main train station, the city center and Hellbrunn.
There is a bus every 15 minutes (20 minutes on weekends).

Thanks Emily. I believe I took a train to a metro and then a metro to my hotel near St. Stephens.