So I'm a 15 year old teenager who moved to Germany like 6 months ago (with my family of course) and I was wondering if I could travel alone back to my hometown in Croatia for a few days with a bus during my upcoming school holidays. Of course I have my grandmother to take care of me there. (The trip would be long around 20 hours so from Freiburg to Vukovar)
You are a citizen of the European Union and as such entitled to freedom of movement. Authorities might be concerned whether you are running away but a simple letter from your parents will convince them you are not. You will find the Documents for minors page useful.
In addition to their own valid travel document (passport or ID card), although not obligatory by law, all minors entering or leaving Germany are advised to carry an authorisation (where possible in the languages of both the home country and the destination country) signed by their parents/legal guardian. The document should show:
that the minor has their permission to travel alone
the contact details of the parents/legal guardian
Although you will not cross Hungary their page just gives more details on what such a letter should contain:
The Hungarian authorities advise including in the declaration of consent the date and place of birth and birth name of the minor(s), accompanying person(s) and parent(s), the minor's travel document number, the purpose and place of the stay abroad, contact details (e.g. country, location, address, telephone number) and direct contact details during the stay abroad (e.g. direct contact details of the hotel, relatives or school).
Other than their own valid travel document, no particular official authorisation is required for minors of any EU country to enter Croatia.
Have fun! Travelling at that age is wonderful, I was 16 when I went out alone the first time, in my case to Israel.
I travelled quite a lot around Europe without adults at that age, but I mostly was with my sister. The biggest practical difficulty when you're travelling on your own, is looking after your possessions, especially (for example) when you're asleep or taking a toilet break. The best way to reduce the risk is to travel light! Remember that other people are trustworthy far more often than not, and that the chance of someone being trustworthy is far greater if you choose their company than if they choose yours. I know it's sexist, but statistically, your wallet or phone is more likely to be pinched by a young single man than by a grandmother: my parents always advised us to choose a train compartment that was already occupied by a grandmother or two.
I don't want to be an alarmist because the chance of bad things happening is very low, but it's as well to have a plan: what are you going to do if you wake up on the bus and your wallet, passport, and phone have vanished? Put them in different places, so that if one goes, you still have the other two. Make sure some key phone numbers are on a scrap of paper separate from your valuables.
And having minimised the risks, enjoy your adventure!