Not sure about "aiming high", I'd recommend going for the smaller companies first. Firstly they aren't buried under several million badly written CVs from students and recent graduates, secondly the work is likely to be much more varied.
At somewhere like Airbus you are likely to see a very small part of the overall engineering process, whilst at (say) Britten-Norman you'd probably in a summer or a year placement see everything from design through production to flight test, marketing, etc...
G