How do I calculate the base and collector current in this circuit?
Unless you are good at iterative math, you'll have to make some simplifying assumptions:


The transistor is on, so its \$V_{BE}\$ is about 0.6V.
The transistor is in saturation, so that its \$V_{CE}\$ is about 0.2V.
Assuming the LED is white, its \$V_F\$ is about 3V.


From these, you can calculate the base current \$I_B=\frac{9\rm{V}-0.6\rm{V}}{220\rm{k\Omega}}\$ and collector current \$I_C=\frac{9\rm{V}-0.2\rm{V}-3\rm{V}}{330\Omega}\$, and then find \$\beta=\frac{I_C}{I_B}\$. If you find a transistor whose datasheet \$\beta_F\$ is larger than this value, then the circuit will work as advertised (once the transistor goes into saturation, its \$\beta\$ will drop to the calculated value). However, if you find a transistor whose datasheet \$\beta_F\$ is less, then it won't work as calculated -- and the collector current will be less, limited to \$I_C=\beta I_B\$.