Of course you should see the Hofburg and Schonbrunn. But if you are fan of castles and fortresses, then you might want to consider these half day/full day excursions easily within reach of Vienna to feel like you've stepped back in time. Several of these are 18th or 19th Century recreations of more ancient buildings, but they still carry the weight of history.
Schloss Laxenburg (https://www.schloss-laxenburg.at/en/) - This was a 18th and 19th century Disneyland for the Habsburgs. While a 14th Century castle exists in the park, the architectural highlight is Franzenburg, a "fake" castle built in the early 1800s. It is absolutely lovely to rent a motorboat (cash only) and glide in front of this very scenic location. The grounds are huge and feature a gorgeous gazebo that Maria Theresa used for card playing. The town of Laxenburg is an ancient market town and features a lovely square, which further features one of my favorite cafes (both for cake and savory foods), Kulhanek's Cafe. The cakes are particularly special. To reach Schloss Laxenburg, take the S-bahn (additional ticket as outside of Vienna) to Moedling. From Moedling, bus 215 takes you directly to Laxenburg. The total journey is between 45-60 minutes. There is an entrance fee to the park. I would allocate 3-4 hours in the park/at the cafe.
Burg Liechtenstein (https://www.burgliechtenstein.eu/en/home.html) - The present castle was built around 1800 but the history of the location dates back to the 12th Century. I suggest pairing this castle with Schloss Laxenburg, as to reach this castle you also need to take the S-bahn to Moedling (additional cost). From the Moedling train station, it is a nice walk through Moedling, through the Vienna Woods to the castle. Once you reach the castle, you will be rewarded with excellent views and photo opportunities.
Kreuzenstein Castle (http://www.kreuzenstein.com/) - Do you watch the Witcher? Maybe this castle also looks familiar as it has been featured in some movies. Anyway, it is another delightful castle built on the foundations of a much older structure. I think this place is just lovely and we really enjoyed having cake at the cafe. Be warned, however, that is quite a hike up on foot. We took a tour and really appreciated entering the rooms. To reach this castle, you can take the S-bahn about 30 minutes to the Leobendorf- Burg Kreuzenstein station. From there, it is a 30-45 minute walk.
Hermesvilla (https://www.wienmuseum.at/hermesvilla_en) - Franz Joseph built this villa as an escape for this wife, Elisabeth, more famously known as Sisi. Now a part of the Vienna Museum, this country house is technically within the city limits of Vienna. The villa is also set in a huge forested area called Lainzer Tiergarten, that includes many hiking trails and animals as well as some small restaurants on the edge of the Vienna Woods. This link for Lainzer Tiergarten has a map - be sure to at least go to the Wiener Blick (Vienna View) for a spectacular look at the city from a hillside (https://www.lainzer-tiergarten.at/anfahrt.html). The website for the villa has good instructions on how to reach this area.
Burgruine Aggstein (https://www.ruineaggstein.at/en/) - Located in the Wachau, this castle ruin is often overlooked for the more popular ruin in Duernstein. The ruin at Aggstein features sweeping views of the Wachau and substantial remaining buildings, making it easy to envision these robber baron's towers. The only trick here is that you really need a rental car to reach the top of the castle, so great if you are making a day trip to the Wachau and already have a car.
Have fun!