What are the best schools for mechanical engineering?

The truth is there are a large number of really good mechanical engineering programs. I graduated from Rose-Hulman Institute of Technology with a Mechanical Engineering degree, and they helped me build a great career. You need to think about a number of factors to pick the right program for you. I've listed a few below. Check out US News engineering rankings to further target the best universities.
1) Would you learn better in a small close class of a few hundred engineers, or a diverse class of many thousand students in different fields? Small specialty college vs. Large university
2) What's your affordability range? Depending on your economic situation, your affordability can widely range. Sometimes having less money makes more colleges affordable due to generous aid programs. Find a degree program that lets you graduate with less debt. That will let you get on with your life faster and enjoy the fact you have a well paying career ahead
3) Look at the colleges graduation placement rate, average salary, and top recruiting companies. Are you likely to get a job in an industry you enjoy? Most of the better colleges have relationships with many large companies. Those companies annually hire interns, and hire many people out of each graduating class. Colleges often tend to get a reputation at being better in certain fields, and a good indicator is who is hiring. I knew people who went into automotive, robotics, manufacturing, and industrial equipment. Start with the end in mind. College campus recruiting offices can direct you to their campus placement office.
4) Some programs are 4 years and some are 5 years. That's a significant amount of money you're out of pocket, or a more difficult 4 years.
5) What is the reputation? Your college program will be on your resume for life, and its reputation will follow you. Be wary of a party school that can turn off a recruiter before they ever talk to you. Having a good reputation locally to where you want to work is important. Most colleges send more people to the surrounding states than to the other side of the country. There are a lot of proud UT and A&M engineers in Austin who love living here, and came from a good engineering program. Local companies gladly recruit from both school as they are likely to get a long term employee.
6) For competitive first jobs, your internships and senior projects will be your differential. As an engineer, favor schools that help you get practical experience before graduating. You'll have more to talk about from those experience during your job interview than about all 4 years of college.
7) It's not all about technical skills when you graduate. What separates you and advances your career has more to do with your writing, speaking, financial acumen, project management, and other softer skills. Value programs that ensure you have enough hours to improve those skills while in college.

8) Make the decision early if you want a masters or direct to the workforce. If you want a graduate degree, you should go to a larger school where you are more likely to get opportunities early to be part of research. Look at the engineering specialty field you want to excel at within Mechanical, and see if most of the people you admire have bachelors or post grad studies. Most of the time, you're fine with just a Bachelor, and some things like robotics may be better off with additional years of research experience in funded labs.

Top 10 Mechanical Engineering Schools in the World
Based on the QS World University Rankings by Subject 2015
Rank
Name of Institution
Location
1 :Massachusetts Institute of Technology (MIT)
United States
2:Stanford University
United States
3:University of Cambridge
United Kingdom
4:University of California, Berkeley (UCB)
United States
5:University of Michigan
United States
6:Imperial College London
United Kingdom
7:Georgia Institute of Technology
United States
8:National University of Singapore (NUS)
Singapore
9:University of Oxford
United Kingdom
10:Harvard University
United States

Webster College, Wabash Indiana, Purdue, South Bend, Ind, any UC system in CA, if out of US, Canadian, U Toronto, my alma ,King's College ,UK

The internet can show you the "best" engineering schools,but I have hired dozens of great engineers over 35 years from both, "well known" schools to almost unknown schools. I would start with trying to stay with an "in-state" university that offers the basic engineering field you are interested in...chemical, mechanical electrical or civil. Paying more for an education will not get you a better job. I found what is more important than where you went to school is how well you did and the knowledge you gained. Work hard, keep up, and try to get an internship before you graduate...these will give you the greatest chance of securing the job you want. Good luck.