I am applying to some PhD programs this year. If I just ask my advisors to choose the school for me, they might think I am lazy and have done no research. However if I do that on my own, I am afraid that maybe the advisors think I am arrogant for aiming for schools out of my reach. The problem is I don't know how strong my profile is and I think that my advisors should be better informed than I am. What should I do?
Fortunately, this isn't an either-or proposition! I agree with your assessment that you shouldn't come to your advisors empty-handed and ask for some schools to apply to. However, it would be perfectly reasonable for you come to them with a couple schools you like - probably spread out by competitiveness and such - and maybe some guidelines for what you want to focus on and where you would be willing to move to.
Ask their opinions on the schools you picked, and ask them if they have some others they would recommend, based on what they know about you.
As @Jeff says, you should follow both of the paths you suggest: develop your own ideas, and then talk to your advisor. (And don't be offended if their perception of your potential versus those programs is not the same as yours. They still may be wrong...)
An important aspect both for you yourself to think about and for discussion with your advisor, is what your ambitions/goals are, and how much risk you are willing to accept, and also stress. That is, there is a genuine and large issue about what your measure of "best" is. It is important also to realize this.
Added: I forgot to mention the practical point that you should contemplate whether you'd prefer (for example) being one of the weaker students at a top place, or one of the strongest students at a less-than-the-absolute-top place. This can have a psychological effect on you while in the program, and can affect your prospects afterward: the second-best student of a famous person in a given year will take a back seat to the best student of that famous person in many (not all) job application situations. (No, this is not sensible, but, in my observation, things tend to work this way mostly. It is harder to justify a job offer that might be accepted, but to the second_best, than to make an offer, that will almost surely be declined, to "the best". The huge point is that the way hiring works (in the U.S., in math at least, at R1's...) is that "the second best" in_a_given_field will never get a chance, because hiring ops rotate through subfields...
But, I think, the hiring game outside of R1's is less nutty... but/and that is a fact that should play a role in your career planning.