I am writing a statement of purpose for Ph.D. applications in the US and Europe. In doing so, I have come across the idea that I should not use first-person pronouns in formal writing. Does this apply to statements of purpose? How do I describe my academic/research/other experiences in the statement then?
It's perfectly normal to use first person pronouns in your statement of purpose. I've never heard of anyone deliberately avoiding them, and frankly I'd imagine it would be pretty bizarre to read!
The statement of purpose is not exactly "formal writing", like an academic paper would be. It's not informal either, it should be written fluently and professionally, but this type of personal essay is perhaps is closer to formal speech (or maybe a formal email) than it is to academic writing.
I suspect your source was talking about using a passive voice and not an active voice (e.g. "X was increased" vs "I increased X). That was standard practice in academic papers and scientific reports in the 20th century. It's sometimes seen as old fashioned now, as the active voice is gradually becoming more common, but both styles have their advocates in academic publications.
In any case, that debate certainly doesn't apply to a personal essay. Use first person pronouns normally in your SOP.
This is more general advice than you ask for, but yes, use the first person as the answer of Gumbercules suggests.
However, you may have the wrong idea about SoP in general. It isn't a document praising yourself for past accomplishments or a recapitulation of your CV. It is a forward looking document giving your plans for the future, both in studies and beyond. It isn't at all about what you have done, but, instead, what you plan to do and how you want to go about it.
So, don't describe your "experiences" there. That is for the CV. And, don't use the SoP to try to explain any past failings.
If you have a subfield in mind or even a research direction you are interested in, put it in the SoP. If you are pointing toward a career in academia, say that. You don't need a formal research plan or topic for most such things though some specificity is helpful, especially if you hold a masters already.
The CV details the past. The SoP looks entirely forward.