How to select resistor values for a current amplifier?
I am using 5 V at the "Trigger" pin, so 1 kOhm at the base would provide ~4.4 mA. Amplifying this would bring me to ~264 mA (with Hfe = 60). Now going through the PNP would amplify this current again to a ridiculously high value.


Your current gain formula only works below transistor saturation. Once it's saturated the current that flows depends on the limit set by the external components. Given that R36 is 1k the max current you can get through Q2 is \$ \frac {V_{IN}}{1k} \$. e.g., For a 12 V supply you can't get more than 12 mA through Q2.

Similarly for Q3, the most collector current you can have is \$ \frac {V_{IN}}{R_{23}} \$.


  What is the correct way to find these resistance values?


Start at the end and work back.


Let's assume that C25 and C11 are charged to Vin and that the SCR trigger voltage is low relative to Vin.
With Q3 hard on we can calculate R23 using \$ R_{23} = \frac {V_{IN}}{I_{TRIG}} = \frac {V_{IN}}{0.2} \$.
Minimum base current to turn Q3 hard on is \$ \frac {I_C}{h_{fe}} = \frac {0.2}{60} = 3.3~mA\$. You should put a good safety factor in here to cover variation in transistors, tolerance of resistors, etc. As discussed above the maximum current is given by by R36 and, since that's a reasonable value (12 mA in our example calculation) let's go with that.
We now need to drive Q2 into saturation so R27 could be calculated the same way. Most of us don't bother and run 1 to 5 mA through the base to be sure to turn it hard on. (If it were a battery circuit you might think differently.)




Caution

You are charging C25 and C11 through R28, 10k. The charge time constant \$ \tau = RC = 10k \times 20u = 200~ms \$. If you give frequent or long pulses on the SCR then voltage on the capacitors will fall and your circuit may fail to operate until you give it a rest.



See also my answer to 
How to calculate currents and voltages with transistors circuit
where the OP was having the similar confusion.