Hello: I am trying to design a circuit in which one LED (D1) turns on after a certain amount of time has passed and then a second LED (D2) turns on after a different amount of time has passed. This is my current circuit and it works well sometimes but seems to falter sometimes. Do I have the correct set up for a my transistor or am I using it in the completely wrong way?
The transistor symbol is incorrect for its part number, so we have no idea how you think the circuit is supposed to work.
It looks like your intent is that Q1 function as an emitter follower to supply power to the U2 circuit after the U1 circuit delay. The problem is a stacking up of voltage drops to the point that U2 cannot run reliably. Look at the output voltage range on the datasheet and see that it is not 6 V when Vcc = 6 V. Then there is the voltage drop across Q1 because it is not operating as a saturated switch.
One possible solution is to change Q1 to a PNP type and drive it with the Discharge output of U1. This should deliver around 5.9 V to U2, enough to operate reliably. The discharge output is open-collector, so there will need to be one base current limiting resistor and one base pull-up resistor to assure complete turn-off.
An alternative is to have both 555's connected directly to Vcc, with different timing component values. This eliminates Q2.