Absolutely okay to ask!! I'm putting together a lecture for the end of April right now actually and the topic is a little out of my wheelhouse, so I'm doing all this stuff right now myself! Including the 20 open tabs....
I would say the process is similar for me in fic vs work, work is just more intense and in-depth and requires more cross-verification. I would love to do more research in a library setting, but the reality is it's just not practical, so it's almost all online for me at this point. I do have a lot of academic reference books that I've bought for my personal use though so I definitely pull those out as needed.
My biggest tip for starting research on anything and how I start most often, especially if I'm not very familiar with the topic in question, is to find the Wikipedia page(s) in question and jump to the bibliography section. That lets you get right into the quality sources without having to hunt from scratch. I'll also just browse the article for a while and then when I see something of interest, I'll check to see what the source was and try to track it down! If it's paywalled, give Sci-Hub or Lib-Gen a try before you pay for anything.
Google itself is fine, but I highly recommend using Google Scholar if you're looking for any real depth on the subject. It works just like a regular search engine, but it ensures that all of your results are academic grade sources. This is usually where I go after I've gleaned all I can from Wikipedia sources.
Usually the final step for me is a dive into JSTOR if I don't have what I need once I've exhausted the Wikipedia and Google Scholar results (usually this is for real, work related research, fic doesn't require this much depth 99% of the time unless you just want the info). I love JSTOR for a lot of reasons, but it's available to anyone and you get like 100 (?) free articles a month even without a .edu email. They also have a great filtering system so it's easy to narrow everything down quickly. Sometimes I'll do the bibliography method here too! Looking at the sources on published academic articles will get you into the real nitty-gritty and I did this a LOT when I was writing my dissertation.