There's lots of ways to start a new lab. Some, however, have proven more successful over time than others. Whereas the responsibilities of different new PIs are wide ranging, most include advancing a research program. How does one do that?
I am a huge fan of taking a diverse approach to one's research question. My lab has had success working across different system and focusing on a couple of main questions, who's trajectories may be headed in distant or similar directions at any one time. I think the key to surviving in a time of tight budgets is flexibility and locking into a single question/system is unlikely to make that simple. Obviously many people find success going down a single rabbit hole, but I don't think I would be as effective at that.
With that said, the new PI cannot chase research projects like butterflies in a garden if the hope is to build a program that will attract external funding. You have two resources as a new PI that you must invest very wisely: Time and Money. They are finite and wasting either has far greater consequences for you that it does for your senior colleagues.
Really cool opportunities will come up all the time. Your ability to discern which ones are worth your resources, and particularly how much of your resources, will play a large part in whether your research program takes off or sputters. As tempting as it is to chase down everything that comes along, spreading yourself too thin is a bad trap to fall into.
Identify your bread and butter and invest heavily in that. Cultivate side projects that you contribute to, but don't do the heavy lifting for. Your grant writing and paper writing should mainly concentrate on advancing your core, with minor contributions to side things. The bar for funding, WRT preliminary data and proof that you know what you're doing is too high to mess around. If your goal is an externally funded research program, having initial focus is fairly critical, before you broaden your scope.