Get yourself a digital voltmeter. Set it to "DC" with a scale of 20 volts or similar--the closest scale value over 15 volts DC. In the ideal world, when you go to start your engine, the starter assembly gets full battery current without any voltage drop along the way. Have someone try to start it when you're making measurements. Don't worry about which lead is positive vs negative. Current needs to go from the TIGHT battery positive to a solenoid somewhere, and return through an equally tight ground path to the tight negative battery terminal.
1) First, measure across the battery. When it is "unloaded", it should measure something on the order of 12 to 14V (if you have the leads reversed, -12V to -14V -- doesn't matter).
2) Attempt to start. If the battery voltage doesn't change much, you don't have a battery problem. If the battery voltage drops considerably; like 10V or less, you have a battery problem.
3) Find the starter/solenoid and measure from the positive terminal to the closest reliable ground point, and try starting. Record that value. It should read within 0.1 or 0.2 volts (DC) of what your first_across_battery measurement was. If you have any drop greater than this, especially a volt or two,
4) Then measure from the positive of the battery to the positive of the starter/solenoid when attempting to start. If this has a greater voltage drop than, say 0.5 or 1.0 Volts, you have a loose positive connection somewhere along the positive path. If this isn't the problem
5) Repeat with the negative of the battery to the closest ground point near the starter/solenoid when attempting to start.
So, around the loop, if all connections are good, you should see something like this when you start: Battery drops a couple of volts max from its charged, no load state. A 0.1 or 0.2 volt drop from positive battery to positive starter/solenoid. A drop from starter/solenoid positive to closest ground very similar to what your battery voltage drop was. And a 0.1 or 0.2 volt drop from the closest starter/solenoid ground to battery negative. Ie, the battery voltage voltage drop = positive (red) wires battery positive to starter/solenoid positive terminal voltage drop plus starter/soleniod voltage drop from positive input to closest ground plus starter/solenoid closest ground voltage drop to negative battery terminal drop. If you always keep the red voltmeter wire closest to the path of positive battery terminal, it will add up (good system) something like: 11.2V = 0.1V + 11.0V + 0.1V under a starting load.
This all should tell you if you have a bad battery, a wiring/connector issue, or a bad solenoid/starter.