A good degree helps.
Assuming that you've recently graduated, the subject matter/industry of whom you're consulting for is not critical and will be developed over time.
Fundamental skills you can build now:
-Problem solving: Not simply answering a question in your head, but rather your ability to coherently string a set of thoughts together and clearly articulating your thought process. Most problems have existing solutions. You are not expected to reinvent the wheel here, but rather take an existing methodology/approach and think of an approach to apply. Case studies help.
-Silver tongue: who are we kidding here? This doesn't apply to your colleagues within the firm, but clients as well. Cover a broad range of topics. My mentor who is one of the most interesting people I know can talk about a range of topics ranging from quantum physics to ancient/modern history, current affairs and financial markets.
-Appreciation for... the finer things.(inc. travel, food and drink): A time will come when someone will seek your recommendation for...something. If they don't ask for it - Give it anyway! and consider it pro-bono! You want to be seen as someone who's well rounded. Considering you will be working around 50-60 hours a week.
I think that's enough for now.