Starting problems

If that's a rather loud clunk once each time you turn the ignition switch to "crank", and it does crank eventually, that's due to worn contacts in the solenoid. They can be replaced separately, but most people just replace the entire starter. It will get progressively worse over the next few months.

DO NOT DISCONNECT THE BATTERY CABLE! You're lucky you have an old reliable car like the old stuff I drive. The battery is the key component in helping the voltage regulator maintain a safe steady system voltage. Had you increased engine speed with the cable disconnected, you would have very likely destroyed the Engine Computer. Newer cars can have dozens of computers and many of them will be damaged when system voltage goes too high. Disconnecting the battery cable was a trick done years ago, before computers and fuel injection, by mechanics who didn't understand how these simple charging systems work. Please don't do that again.

It sounds like you just have a bad battery. Use an inexpensive digital voltmeter to measure its voltage. A good, fully charged battery will read at least 12.6 volts. A good but discharged battery will read closer to 12.0 volts. If you find 11.0 volts or less, a cell is shorted and the battery must be replaced.

If you want to test the charging system, measure the battery voltage again while the engine is running. It must be between 13.75 and 14.75 volts. If it stays near 12.0 to 12.6 volts, the system can be diagnosed by taking the voltage readings on the two small wires on the back of the alternator.