Once I install the new solenoid, DO I just press the button and see if the starter motor clicks?
Solenoids click, motors spin. The purpose of a starter solenoid is to transmit a small amount of power from a switch to a large amount of power to a the starter motor that requires high power to operate. A starter solenoid is basically a switch that is in series with the ignition switch.
Unless you have a lot of money to pay others to fix your Healey, you should research basic automotive systems so you can better analyze symptoms to come up with a diagnosis.
For example this is from wikapedia:
[edit]Automobile starter solenoid
Main article: Starter solenoid
In a car or truck, the starter solenoid is part of an automobile starting system. The starter solenoid receives a large electric current from the car battery and a small electric current from the ignition switch. When the ignition switch is turned on (i.e. when the key is turned to start the car), the small electric current forces the starter solenoid to close a pair of heavy contacts, thus relaying the large electric current to the starter motor.
Starter solenoids can also be built into the starter itself, often visible on the outside of the starter. If a starter solenoid receives insufficient power from the battery, it will fail to start the motor, and may produce a rapid 'clicking' or 'clacking' sound. This can be caused by a low or dead battery, by corroded or loose connections in the cable, or by a broken or damaged positive (red) cable from the battery. Any of these will result in some power to the solenoid, but not enough to hold the heavy contacts closed, so the starter motor itself never spins, and the engine does not start.