What does it take to be an aeronautical engineer? What is it like?

A career in aerospace engineering can mean so many different interesting and exciting things!

First of all, you don't need to have an aerospace engineering degree to have an engineering career in aerospace. I have lots of colleagues with aerospace engineering degrees who describe an aerospace engineering as basically a mechanical engineering degree plus more fluid dynamics courses. However, not all employers seem to know this, and if you decided to do something outside of aerospace, some companies would tend to choose mechanical engineers over aerospace engineers, even though the academic background is so similar for the two. In addition, the proportion of engineering jobs at an aerospace company that require an aerospace engineering degree is actually fairly small. So many jobs can be filled by other types of engineers, and there are quite a number of aerospace engineering jobs (such as in avionics and controls) where another degree (like electrical engineering) is preferred over an aerospace engineering degree. There are important exceptions: an aerospace engineering degree is needed or much preferred for jobs that involve fluid dynamics and aircraft/engine performance. You might consider pursuing mechanical engineering and specializing in aerospace engineering later when you have a better idea of what you want to do. (I should probably note/disclose that I have a BS in Mechanical Engineering and I have been working in the jet engine industry for close to 15 years.)

Back to your main question about what a career in aerospace engineering is like: there are so many different types of engineering jobs in aerospace. Here are some examples, roughly following the life cycle of a commercial aerospace product as a guide:

When an aircraft, engine, system, or component is first being developed, design or system engineers define the high-level characteristics of the product. This involves a lot of analysis and computer-aided design done at a desk and in teams, but usually not too much "hands-on" work. For a commercial aircraft, this would entail defining number of passengers, aircraft mission (where it would go and how far), speed, thrust requirements, cabin environment requirements, wingspan, etc.

Then comes the detailed design phase, where the aircraft and components are defined down to the hardware level. Design engineers use 3D models to define wing shapes, landing gear components, cockpit configuration, etc. This is done in concert with other disciplines, including structures engineers (who model the stresses in the designed parts to make sure they won't fail), materials science engineers (who analyze and select the proper metal alloys, coatings, and material processing methods for the parts), and manufacturing engineers (who determine the best way for the parts to be manufactured, what manufacturing tooling is required, etc.). Throughout this design phase and other phases, project engineers will take on the management and coordination of these tasks. They need to have a good understanding of what each group does, and they need to get everyone together to agree on the product design while staying on schedule.

Then prototype parts and systems get made and tested. This involves a lot of hands-on activity by engineers. The manufacturing engineers implement the processes defined above to convert hunks of material into parts. These parts get tested by test engineers who stress the parts and systems to verify the accuracy of the computer modeling that was done previously. The design engineers from above take the results of this testing and refine and optimize their designs.

This process culminates in a final product that is provided to the customer, such as an airline. The airline will often have its own airframe and powerplant engineers that will monitor aircraft and system performance and help with troubleshooting when there are "mechanical difficulties" that cause delays and cancellations. These airline engineers will also work with the manufacturers to improve the product and the maintenance practices for the aircraft systems. This part of the business is often very hands-on, dealing with messy hardware problems and finding ways to fix them. Similarly hands-on is the huge aftermarket business in commercial aerospace. Repair engineers find and develop ingenious ways to fix parts with advanced welding, coating, and other repairs.

In summary, if you like aviation and you like engineering, there is definitely something for you. You can do advanced analysis, modeling, and number crunching every day as a structures or heat transfer engineer. Or, you can get your hands dirty as a manufacturing or test engineer. Or, you can manage projects as a project engineer. Plus, if you ever want to move on from engineering, there are plenty of other employers who love people with engineering backgrounds because they know how to make decisions based on data, which is something that is hugely important in any business.

As engineering goes, Aerospace Engineering is a specialization compared to other more general fields of engineering. That means there are fewer jobs than say an electrical or mechanical engineer.
Depending on your interests a career in this field might involve aircraft design and performance of anything from Cessnas to military jets, flight control systems and the software which models it, flight simulation which is heavily software-based, or even space flight design and operations (which is what I do). Regardless of the area, there will be a heavy computer simulation component and software skills are valuable.
In an aerospace field, the work can be highly specific to a particular aerospace system so expect a lot of training once you landed a job. Your college degree is the foundation but it only gets you in the door. You'll be learning systems that are not described in any textbook. People that can eagerly grasp new concepts and will do well.