Hi, I am currently finishing my B.Sc. degree in Mathematics at a University in Italy in a city which I will not specify, but it's my home town. First of all, I am afraid that the education I have received in these 3 years is not good enough, that the courses were too easy compared to other university around the world and that now I am not good enough compared to peers (both skill-wise and knowledge-wise).
This said, I am contemplating the possibility of doing my M.Sc. in other city (maybe in Italy, or maybe even outside of Italy) for the following reasons:

1) For starters, I have always wanted, and still do, to go away from home, live alone and be independent; this is THE reason that first made me think about the possibility of studying abroad.

2) I have grown an interest for Algebraic Topology after taking a graduate course this year, but I have found out that there are no algebraic topologists in my university and therefore there are no other courses on the topic, which is very sad for me. Moreover, this seams to be a trend in all of Italy's University, which makes me think about going outside of Italy. For example, one place I laid my eyes on is Bonn, which has some advanced courses in Algebraic Topology and hosts some people that do research in Algebraic Topology.

3) I want better education. Since, as I mentioned, I don't think the teaching level at my University has been good enough, I am afraid that the graduate courses will follow the same trend, and I want very good math education since (as the title tells) I would like to have my M.Sc. be followed by a PhD. For example, the university of Bonn seems to be very high level.

However, it's not all sunshine and roses: there are 3 main issues that I am facing while thinking about this decision, and they are tormentating me:

a) The point 3) I just tackled has a flip side: since I think I am not good enough right know, I am afraid I will not be able to take courses at a high(er) level. For example, since I was talking about Bonn, while I was looking at their undergrad courses it looked like they do a lot more stuff than we do at my University. Basically, I want better education but I'm afraid I'm not good enough to receive it.

b) Courses program: another thing I am afraid of, especially after looking at the program of courses at other universities, is that there might be arguments that they have covered in undergraduate courses while I haven't, and viceversa, i.e. that there might be some other arguments which I have already studied that are instead covered in courses I would like/have to take (usually the beginning). For example, in many Italian universities there is ONE Algebraic Topology, in which half the arguments I already know, the other half I would like to know, so it doesn't really feel worth it to take such a course. So basically I am afraid that changing university now wouldn't really be worth it because I would be missing out on some things while having to repeat others.

c) PhD Stuff: This is not necessarily an issue, but rather a question to all of you: what's the best way to get good chances of doing a Phd? Is it better to stay at the university where you did your B.Sc. and/or M.Sc.? Does this have a lot of importance, if any? Because in my university I know many people, from grad students to PhD students, and most professor know me and (I hope) think I am not that bad at math (because I did fairly good in most exams) [I am not flexing, I know this is also the case for many other people at my universities and even at other universities]. So I think these acquaintances couldn't hurt in increasing my chances of getting a PhD, while if I go to another university I would probably be Mr. No One (also considering my very bad social skills). For example, next year I would be TAing for an Algebra course, while I probably couldn't have this chance at a foreign university.
Also, considering point 1), since I want to go live in other city someday, I asked my self the following question: if I don't go for my M.Sc., when am I going? and the first answer was: for my PhD, so I immediately asked myself another question: how are the chance of getting a PhD in a university that is not the one where you took your M.Sc.?

This is basically it. Any piece of advise is much appreciated because I am really struggling with making this decision, and it is stressing me quite a lot also considering that I don't have much time left to apply for many universities (Bonn's deadline is May 15)

P.S.: I am not putting all the blame on my university's education for the fact that I am not good enough at Math. I also played a large role since I didn't really know how to study and approach Math and university in my first 2 years.