In your early years as an engineer, you may well get your hands dirty testing things, however the money stops going up after a few years and most engineers end up managing engineering projects or just moving out of the sector and into standard management. I believe the statistic is that 80% of companies have engineers on their board of directors, so you see many/most leave the industry after a while.
As for the degree itself, you will have to do practical work, but also a lot of book learning (hence engineering having the most contact hours of just about any course).
The sort of role you might have differs depending on where you choose to work, but it could be something like receiving the specification for an engine cylinder head and being asked to come up with some designs and run simulations to see which is most effective.
Or, you could be testing lubricants and doing a variety of destructive tests to see how they perform.
Or you could be a race engineer for an F1 team.
Seriously the brilliant thing about engineering is there are tons of options in the field itself, and you can easily move off into finance/management or whatever.
If you want hands on, it's yours, if you don't then that is doable too.
Anything you wan't me to be more specific about?
EDIT: I should make it clear that a Mechanical Engineer and a Mechanic are NOTHING alike. An engineer designs and tests, and a mechanic merely fixes things when they go wrong. You can get your hands dirty as an engineer in some ways but generally you shouldn't think of it as an oily boiler suit sort of thing. Unless you are in F1.