Definitely have the battery checked out first. If it checks good, be prepared to do some voltage drop testing to find any bad connections or bad cables.
1.) Will the car start if it is jumped? Then clean battery terminals and check battery for low charge and dead cells. A good battery will measure 12-13 volts at full charge with the ignition switch in the Run position but without the engine running.
A voltmeter placed across the battery terminals should show a minimum of 9.5-10 volts when the ignition switch is turned to the Start position and the starter engages or tries to engage. Less than this will result in a clicking solenoid, or slow cranking (if it cranks at all) or a starter solenoid that sticks and welds the contacts together.
Most auto parts stores will check your battery for free. It does not have to be installed in the car to have it checked; you can carry it with you to the auto parts store.
The battery posts and inside of the battery post terminals should be scraped clean with a knife or battery post cleaner tool. This little trick will fix a surprising number of no start problems.
The clamp on with 2 bolts battery terminal ends are a know problem causer. Any place you see green on a copper wire is corrosion. Corrosion gets in the clamped joint and works its way up the wire under the insulation. Corroded connections do not conduct electricity well. Avoid them like the plague...
A voltmeter is handy if you are familiar with how to use it to find bad connections. Measure the voltage drop across a connection while trying to start the car: more than .5 volts across a connection indicates a problem.
See Automotive Test Tools for help for help troubleshooting voltage drops across connections and components. .
2.) Check the battery to engine block ground down near the oil filter, and the ground behind the engine to the firewall. All grounds should be clean and shiny. Use some sandpaper to clean them up.
3.) Jump the big terminals on the starter solenoid next to the battery with a screwdriver - watch out for the sparks! If the engine cranks, the starter and power wiring is good. The starter relay is also known as a starter solenoid.