Namely, I would like to look in my room Netflix and unfortunately because my Wi-Fi at home is too weak for it, I do not know now whether I should rather get an Internet stick or a modem box? What do you think or are there better alternatives?
With an internet stick you would have a separate internet connection, it will not take you further.
What you are looking for is a "Wlan Repeater"
I've already heard of that but my Wi-Fi is generally slow so I wanted my own internet connection. Does an internet stick cost a lot?
Stick yourself not expensive, often give the network providers for free. You should have fast and stable mobile reception and find a suitable data package.
Unlimited mobile internet is still quite expensive, but with a small data package you can't watch a lot of Netflix. Very rough, but you can expect about 500Mb per movie (of course not in FullHD).
If you have good reception in the Vodafone network, I would recommend trying out Gigacube from Vodafone with 200Gb, they can also test for 30 days.
The best would still be the existing Internet connection faster and make Wlan stronger.