I am assuming you are an undergraduate; correct me if I'm wrong and I'll amend my answer.
It has been my experience that summer internships at a university that you don't attend are very rare. My advisor explicitly ignored all cold emails from students asking if they could work with her during the summer (even if they were not asking to be paid). Logistically, without being enrolled in the school, things like getting access to school servers, badgeing/keys for offices, etc. is just that much more difficult, and professors just don't have the time to vet students from external schools, nor much of an incentive to do so.
With that caveat:
Ask home university/department/professor for contacts and use that connection?
Absolutely, but it is going to take more than getting contacts to get a summer internship. Unless your professor reaches out to those contacts himself/herself, you're probably not going to get much traction.
What if I have no prior research experience and professors only know me from classes?
Then you are less likely to entice them to help you. You're better off trying to work with those professors at your own university instead of trying to use them to work somewhere else.
Google and send CV + Statement to (many) professors via email and hope for the best
Please don't bother them -- they get too much spam to begin with. You'll not only waste your time, but theirs as well.
Google for positions that are explicitly offered for 8-10 week summer internships
This is the best option, but I suggest not limiting yourself to universities and to branch out to look for internship experience in industry (if this is such a thing in your field). For one, there are probably more opportunities, and secondly, they are generally more structured and you might get more out of the experience.
Incidentally, many professors use the summer months for work-related travel and to take vacation. This makes it even more unlikely that you will find a position, as the number of professors around during the summer is minimal. On the other hand, if you happen to find a professor that will assign you to one of his/her graduate students, you could potentially still work in the lab under the grad student tutelage instead of directly for the professor (and, indeed, this is probably what will happen in any lab -- as an undergraduate you will probably work more for the graduate students than for the professor, although this obviously varies by school and professor).
