How can I use a PNP or NPN transistor as a switch?

Get a basic op-amp and use it as a comparator

http://www.bcae1.com/opamp.htm

Use an NPN transistor (of adequate current rating) to "ground" the -ve lead of the relay coil, the other end connected to +ve power.

In the base, use a 4.7K 1/2W resistor.

To eliminate the 2V, use 3 or 4 Si signal diodes in series, cathodes towards the transistor base. Each diode will drop the voltage approxomately 0.6V.