What problems did Arabs and Israelis face with the independence of Palestine?
This question is a little hard to understand. You say "did" but Palestine is not independent (if you're talking about the Palestinian state that is supposed to get made out of the West Bank and Gaza). Israel did become independent, but it was not called Palestine after independence...
I'm going to assume that you're talking about the time when Israel became a country in 1948. At that point, the Israelis faced the problem of being ringed with countries that hated them. Various Arab countries attacked Israel as soon as it became independent. Because it was ringed by enemies, Israel was ina precarious situation.
The Palestinians faced the problem of having been driven off their land. Most of them had chosen not to become Israelis. They had to figure out how and where they would live now that their land had become part of Israel.
Since then, of course, the two groups of people have continued to face these problems which still exist to this day.