There are a variety of summer papyrology institutes.

That being said, you don't mention what kind of education you have. I'll assume you have a bachelor's degree in history. If you want to be a papyrologist, you'll probably need at least an masters', but probably preferably a PhD in history, classics/ancient near east, or archaeology.

You'll also need to know probably at least two dead languages and at least one non-English modern language used by scholars in your field. Probably you'll want to learn German, French, or Italian. You may end up specializing in a particular region, which will influence the languages you learn. If you want to do stuff with Greek papyri, you'll want to learn German or maybe French, as well as Attic Greek and probably Latin as well. If you want to work with Egyptian papyri, probably you'll want to learn French, Egyptian, and Greek.

I would imagine any graduate program you'd be enrolled in would help you to learn these. However, some graduate schools require some language knowledge beforehand (I think this is more common for history and classics than for archaeology and the near east). For example, the classics program at UW Madison requires the linguistic equivalent of a BA in Classics, which they define as three years each of Greek and Latin.

I am not an expert on papyrology, so I cannot tell you about the career path and pay, but I would imagine papyrologists have a similar career as epigraphers or other specialists within the fields of history/archaeology. From my experience with classicists (I spent a year studying classics in Athens), probably your best bet is to get a PhD from someplace with a strong classics/archaeology program, and make a point of taking courses on papyrology and getting funding to go to summer institutes.

Try to do either PhD research or a postdoc abroad in situ, with some sort of program like ASCSA. Ideally, you could find a PhD advisor who studies papyrology. I would image you could also look on the website of the American Society of Papyrologists and perhaps even email them and ask for their advice, as well. The goal would probably be to get a professorship and focus your research on papyrology. The more you get involved with the field, the more recognition and credibility you'll get.

I hope this was helpful--as I said, I'm not a papyrologist, but I've spent a lot of time around professors in general and classicists in particular, so I feel like I understand this well enough to offer some suggestions.