Time series analysis. Autocorrelation and differences

I try to analyze the Wolf numbers. I have never analyzed time series before. Firsty I calculate autocorrelation coefficients. Biggest coefficients were for delays: 1, 2, 6, 7, 8, 9, 10. I calculated differences D(-1), D(-9). But for both calculated time series autocorrealtion coefficients for delays 1 and 9 were big and significant. My (first) question why it so? Is it normal situation?
Yes, this is normal. Differencing a time series helps with making it stationary and susceptible to subsequent modeling. It does not remove serial correlation though. To capture serial correlation, you should consider an ARMA(p,q) model as applied to the differenced time series. To identify the optimal specification (including parameters p and q and potential "seasonal" components), you can exploit the Box-Jenkins approach. This approach makes heavy use of autocorrelation function and partial autocorrelation function, which you have generated already.