"How do you handle being given a dataset, but no clear objective?"
This will be common.
Apart from the advice above, understand that it is essential to understand the goals of the business you are in, and of your immediate client. Frequently you will need to understand the specific problem that made them turn to data better than they do.
It is very highly common to be presented with data and an unclear objective from your internal or external client - it will be usually your task to supply a goal that can be achieved with the data and will solve the client's actual business problem. An amount of lateral thinking will be required to make the data outcome and business solution match.
I would summarise the above as 'defining the objective is too important (and possibly too difficult!) to be left to the client (alone)'.
In the machine learning context, CRISP-DM is a methodology which tries to solve this problem by iterating through a loop so that additional data understanding can be used in discussion with the client to better understand the original problem. So, for example, they may state a ill-defined goal, a second discussion after you've done some EDA will sharped it a little. When you later produce a model that works well, but isn't on quite the right target, you'll get closer to the real business goal again.
In other words, don't be too disturbed by the fuzziness of the task. Expect to encounter a vaccuum, and fill it to your advantage.
It's a slight sideways shift, but the six sigma methodology attempts to solve this problem in a different context with the DMAIC system (the 'D' standing for 'Define', in terms of the 'voice of the customer'), so it is probable that some tips can be gleaned in resources for the six sigma context (e.g. exercises you can do with a client that help them express what you want more clearly)