So, the IS curve is a set of equilibria: all combinations of income and interest rate that achieve macroeconomic equilibrium are represented by the IS curve. After all, that's why they call it IS: Investment = Savings!!!
Hence, let's take off from here:
Investment = Savings = Public Sector Savings + Private Sector Savings
What's Public savings? Remember: savings is all income left after expenses. Thus:
T - G = Public Savings: Government's income (taxes) minus expenses (gov' purchases, G).
What's Private savings? Well, the people earned an income (Y), paid taxes on it (T), and spent on their consumption (C): Y - T - C
There we have it: Savings = S = (Y-T-C) + (T-G) = Private + Public savings
Question: how do we know S = I?
Put together your first 3 equations, and let's do one bold assumption: Y = Z (i.e., actual expenditure = planned expenditure. Not that bold, is it?)
Y = C_0 + C_1*(Y−T) + I_0 - I_1*r + G
Solving for this equation (beware: Y appears on both sides of it, which means you need to to solve for it), goes like this:
Y - C_1*Y = C_0 - C_1*T + I_0 - I_1*r + G
Y*[1 - C_1] = C_0 - C_1*T + I_0 - I_1*r + G
Y = [C_0 - C_1T + I_0 - I_1r + G]/[1 - C_1]
And that's your IS curve, exhibiting a negative relationship between your two key variables, income and interest rate.