I have been living in Heidelberg for two months now. And although that is nearly not enough to know a city inside-out, I already know my favourite spots, and those I am eager to recommend if you ever have a chance to visit this wonderful, lively city.
1. The castle
This one is a bit obvious, but bear with me - being in Heidelberg and not visiting the castle is a bit like going to Rome and not seeing the Pope, as the popular saying says. This is not a castle per se - it is actually a baroque ruin, which is the symbol of the city. The climb up is a bit tough on the lungs, but there is a Bergbahn that takes you straight to the top, which has one of the best views over the city (just perfect for sunsets).
Besides being beautiful, it is the perfect place for the lovers of history. It was the residence of the Prince Electors until the succession war in the Palatinate. Through here passed by notable figures of the history of the Holy Roman Empire, such as Frederick the 5th. Although the guided tour implies an extra fee, I would definitely recommend it for those who are interested in actually knowing the history behind these walls.
The ticket includes the Bergbahn (both the journey up and down), the entry in the Castle's courtyard, the Museum of German Pharmacy and the caves in which you can find the famous Heidelberg Tun, a giant barrel very famous throughout the world.
Tough to sum up the Altstadt in one single point, but I believe that to truly know a city, one must walk it. If you have the chance, I recommend you to book a free walking tour - if you're not familiar with the concept, you are the one who sets the price on the Guide's work. In my travel experience, being accompanied by someone with general and specific knowledge of a city allows you to know little facts and details that would have been very hard to realize on your own, making the trip way more enjoyable. How else would I know the fascinating story behind why there is a brezel drawn on the wall of the most famous church in the city?
I would recommend starting in Bismarckplatz, where all the transportation meets. From here, walk the Hauptstraße - this is the main street of the city, and the hotspot of commerce in town. This street goes all the way from Bismarckplatz to Karlstor, basically crossing the oldtown from one point to another - you will have to make detours to visit the Universitätsplatz, the Marktplatz and the Alte Brücke.
In the Marktplatz, you will find the Heiliggeistkirche, a gothic church where once stood a wall that divided catholics and protestants. This was where the University of Heidelberg and the Palatine Library were first founded. The building of the Alte Brücke was ordered by Karl Theodor, and therefore still holds his name until today. There are three statues on the Bridge - one of Karl Theodor, naturally, another of the roman goddess Minerva and one of... A Monkey? The reason why the Brückenaffe exists is not certain, however, tourists gather to rub its bronze in order to either return to the city, become rich or have many children.
3. Walk the Way of the Philosophers
This is something I have to admit I haven't done myself, but I definitely encourage you to do it if you visit during days of good weather. The Philosophenweg is one of the most famous walks in town. Although the climb is costly, the view over the city, the river and the castle is the one you usually see in postcards. If you love views, make sure not to miss it.
And that is pretty much it! Heidelberg is a small town, compared to others, but it has a lot of history and fun things to offer. If you do visit Heidelberg, one day, be careful - just as famous artists and poets you once did, you might end up leaving your heart behind.