Mechanical engineering is a broad field that can be applied to many different types of products, but in general, you'd be doing a lot of the same kinds of things.
If you were designing a robot for example, you might start with coming up with ideas for how the arms would work, where to place the joints, the motors that control them, that sort of thing. You would develop 3-dimensional solid models of the design using various software tools like AutoCad or ProEngineer. You could simulate the motion of the arm to see that it moves how you want it to, and change the design as you need to.
You might perform a mechanical stress analysis using finite element modeling to see where the mechanical stresses are in the various structural components and change the design if parts are too weak or too strong. You might also perform thermal analysis of heat buildup in the electronics (with some help from electrical engineers) to see how hot various components are getting, and whether forced cooling (e.g. fans) might be needed for cooling. And if so, you'd participate in the selection of the fans, and their location and packaging
You would select / specify the various mechanical components for the robot arm, such as bearings, fasteners, etc. For components that need to be fabricated, you would select the materials that they would be made from and generate detailed drawings or CAD files including all dimensions and tolerances that would allow machinists to create the parts.
And, you'd participate in the final assembly, test and debug. When everything comes together, does it work like it's supposed to? Probably not the first time. So you'll need to figure out how to fix whatever problems arise, and change the design - usually very quickly as there's now schedule pressure to get the product working and shipped!
In some designs, precision is important, and careful design and tolerancing is called for. In others the design needs to be very stable over temperature so that you need to carefully consider thermal expansion. And in many designs, both are necessary. For example, think about the design of a lens for a digital SLR camera. The individual optical elements inside need to be positioned very accurately to keep the lens focussed. They can't move around when the temperature changes or the optical focus gets degraded. Lots of other precision components inside the lens too, autofocus sensors, motor drive, etc. The design needs to be rugged, compact, easy to manufacture and low cost. And as a mechancial engineer, you have to figure out how to do it! (But you'll have plenty of help, especially just starting out)
This give you some idea of what an ME might do depending on the types of products you might be working with. Over time, you might decide that you like the the analysis aspects and become more of an expert in stress analysis. Or, if you're very creative, you might do more conceptual work, envisioning new mechanisms and structures, etc and let other engineers do more of the detailed analysis and fabrication work. Or, you might develop special skills and expertise in precision design, or adhesives, etc.. and end up doing more work in that area.
Over time, depending on what field you end up in, you'll generally also have the opportunity to learn about the other engineering dicsciplines and technologies associated with the products your working on, and beome more versatile.
Hope this helps and gives you some perspective,
-Guru