A hard start

I'm assuming just by the fact you had a new battery and new starter that sometimes when you turn the key to the start position , nothing happens . Is your vehicle a automatic transmission ? When it doesn't start have checked for battery voltage at the starter solenoid S terminal ? Is there B+ voltage at pins 30 & 86 for the starter relay ? Pin 86 while cranking the engine or trying to crank the engine over . Pin 30 check with key on . Try moving gear selector to neutral . Videos on youtube for basic automotive electrical testing . How to voltage drop test starter motor circuit Find a wiring diagram at www.bbbind.com
Starting System Circuit Description
Voltage is applied at all times to the ignition switch from the IGN A fuse 6 through CKT 242 (RED). When the ignition switch is turned to the START position, voltage is applied to the CRANK fuse 8 through CKT 5 (YEL). From the CRANK fuse 8, voltage is either applied to the clutch pedal position switch (M/T) or the transmission range switch (A/T) through CKT 806 (PPL). When either the clutch is disengaged (M/T) or the transmission is in park or neutral (A/T), voltage is applied to the coil of the starter relay through CKT 1035 (PPL/WHT). Since the starter relay is permanently grounded at ground G105 through CKT 150 (BLK), the starter relay energizes.
Voltage is applied at all times to the starter relay contacts from IGN A fuse 6 through CKT 242 (RED). When the starter relay energizes, the starter relay contacts close, and voltage is applied to the starter motor solenoid. Since the starter motor solenoid is permanently case grounded, the starter motor solenoid will energize two coils. The pull-in winding coil energizes in order to pull the starter motor solenoid contacts closed. When the contacts close, a plunger on the contacts causes the pull-in winding coil circuit to open. The hold-in winding coil then holds the starter motor solenoid contacts closed. Voltage is then applied to the starter motor from the battery through CKT 1 (BLK) and the closed contacts of the starter motor solenoid. Since the starter motor is also permanently case grounded, the starter motor will run until the ignition switch is moved out of the START position. When this happens, a spring in the starter motor solenoid moves the starter motor solenoid contacts and the plunger back to the rest position.

Does starter click? Try turning on headlamps and get someone to watch as you try to start car. If headlamps dim, likely bad starter or battery cable(s). If they don't dim, bad solenoid, starter switch, neutral safety switch. Try starting in neutral and jiggle shifter while trying. If it starts, replace neutral safety switch.

can you jump power to the solenoid to verify its operation?

I will try to help, powell. Your start circuit from battery to ignition switch to starter-and back to battery. You may need to check every thing here, including the grounds, for a loose connection, or just possibly a failing part.
If only complaint is intermittent starter action, and no other problems with ignition switch, let's assume switch is good, so battery to ignition switch is good. Now from the switch, a yellow start wire goes to the instrument panel fuse block, to "crank" fuse #8 (10 amp), from there a purple wire goes to the safety switch. If manual it is a clutch pedal position switch (simple on/off switch-when clutch down switch is on). If automatic, purple wire goes to Transmission Range Switch on the transm.-in park or neutral the purple wire has continuity in and out of switch.
From that switch, the purple wire goes to the starter relay (under hood fuse-relay center). Now the relay: purple wire in start will energize the relay- wire goes to coil side of relay, and then coil side goes to ground. On relay power side, the power feed is a red wire-and it is the same red wire that initially sends power to the ignition switch. A 40 amp fuse in fuse-relay center sends power to ignition switch, and a splice goes to the starter relay as the power feed for relay. When relay is energized, contacts close, and power is sent from relay to the starter solenoid, again on a purple wire. When solenoid contacts close, starter motor is connected to the big battery cable on the starter.

There is your start circuit. try a new or different relay, try a new or test the clutch switch (or transm. range switch). Buy a $5 test light and use it to check where power is lost when starter won't operate. Hold key in start and check if fuse #8 is getting power. Check the safety switch, with key held in start, the purple wire should be hot. Check if relay has a power feed (pull relay out and check terminals) and if relay coil has a power signal in start. Check the relay's coil side ground. If that ground is intermittent, relay won't work.
When it won't start, use a test light, you should be able to find it.
Good luck.

Check the distributor

Start or crank? I bet the answer is crank. It's the solenoid or the ignition switch.disconnect battery, then remove the 2 small wires from starter solinoid. if it still does it it is a bad solinoid or ignition switch. check solinoid first.

I would suspect the starter.

I'm assuming that by turning over, you mean that the engine fails to turn with the starter . Check your battery for 12v and make sure all connections are clean and secure. If battery has less than 12v charge it and check the charging system after re-starting. Lights and radio do not pull the same amperage as the starter and may not tell you much about battery condition. If you have full battery voltage, check the starter for 12v at the battery connection and the small (trigger wire) for 12v with the key turned to start position. If you have both, likely starter has failed.

ONE OF THE WIRES ON THE STARTER IS ON THE WRONG TERMINAL. OR TWO OF THE SMALL WIRES ARE MAKING CONTACT WITH EACH OTHER. CHECK OUT ALL THE WIRES. HOPE THIS HELPS YOU. PLEASE KEEP ME POSTED. THANKS FOR AN HONEST RATING AND FOR USING FIX-YA. GOOD LUCK

test the battery first then go through the ground wires, there will be a few and if one of them have corrosion or are not connected you will not get full voltage to the starter,