I will be travelling from South Africa to Hamburg Germany in September with my 70 year old mother. We will be spending 3 days in Hamburg and then have 9 days available for sightseeing. We would like to visit Innsbruck and perhaps include Prague in our trip . We are nature lovers on a tight budget. Any suggestions would be appreciated.
Long travel distances. Flying in to Munich or Nuremberg would shorten travel times.
I am currently working on my travel plans to Germany in a couple of months and saw your post. Hamburg is a long distance to southern Germany and since you mentioned that you and your mother were nature lovers I would suggest visiting Bavaria and the surrounding area. I went to Prague in 2014. We flew into Munich and then took the DB bus from the train station to Prague. Train travel in the Czech Republic is not the same as traveling in Germany. I don't recommend you try and go to Prague. It is a beautiful city but it's wall to wall people no matter when you go. If you do decide to go, it would be better to fly to Prague from Hamburg and then travel to Germany. Also, I don't recommend going to Innsbruck. Again, a big city and I think your time is better spent exploring southern Germany and the scenic parts of Austria.
With 9 days for sightseeing you can go directly to Munich from Hamburg (6 hours on the fast train) and begin there, or break your journey in Nuremberg (5 hours) which can be quite charming too.
You don't mention where you are flying home from and that will make a difference as to whether you want a one way pathway or a circle to get back to the departure point.
Or you could fly Hamburg direct to Innsbruck, although I don't know the costs.
Munich to Innsbruck is 2 hours, and is a lovely train trip. If you are nature lovers the Innsbruck area has several easily accessible valleys with good bus service or local train service that makes heading out of Innsbruck particularly easy. There are lots of Pensions in the Stubai and Zillertal Valleys which get most of their business from summer hikers or skiers, so you should be able to find reasonable prices in September.
https://www.zillertal.at/en.html
You can also go Munich to the Garmisch-Partenkirchen area and take the cog railway up the Zugspitze, or hike the Partnachklamm. It is 90 minutes to Innsbruck the next day from G-P.
https://www.gapa.de/en/Your-GaPa-experience/Partnach-gorge
https://www.gapa.de/en/Your-GaPa-experience/Zugspitze
I have to agree with others that the 4 hour bus trip to Prague (from either Nuremberg or Munich) will take full day of travel there and back rather than sightseeing.