Transistor Based NOT Gate Not Working

A 10 K ohm resistor, as a collector resistor, with a 0.7 Volt drop VCE, with a 3.0 Volt source, will result in a 230 microamp flow, aka 0.23 milliamps. With a hfe gain of 100 as ideal, that's 2.3 milliamps at best VCE. But a 10 K ohm resistor as R2 would only allow a few micro amps through a led, not lighting it up. Try replacing both R and R2 with larger resistors and it will work. Try 470 to 1000 ohm.

CMOS parts like the 4011 have a very high input impedance and will have a random value if left unconnected.

If you are connecting your switches between an input pin and the positive supply, you need a pull-down resistor (10K or so will do) to ensure that the input will be low when the switch is open. If your switches are between the input pin and ground, use a pull-up to the positive supply to make the inputs High when the switch is open.

Also, any unused inputs must be connected to either the positive supply or ground to ensure they don't sit at a "maybe" state and cause the input circuit of the gate to draw excessive current.

You don't need any resistors between the gates.

As one comment says, it is good practice to have 0.1 uF bypass capacitors between the power supply and ground at each IC - more important with flip-flops and other clocked logic than with simple gates, but it is good to get in the habit...

^{simulate this circuit – Schematic created using CircuitLab}