Turn key to start and nothing happens. All lights are on, battery good. Doesn't turn over---nothing. Switch or starter or some kind of relay??? HELP!!!

If by lights, you mean the dash lights come on, then ignition switch must be getting power. If dash is not working with key on, check the 30 amp fuse link which powers the ignition switch.
If switch is good and starter is good, the start wire from switch goes to, if automatic, to inhibitor relay (the park/neutral function) and then to starter. There is also an inhibitor switch on the shifter or on the transaxle that turns on the inhibitor relay when key is on and shifter in park or neutral. The inhibitor switch is powered by a 10 amp fuse D.
If you have a manual transmission, the start wire from ignition switch goes to the clutch interlock relay and then to the starter. This relay is turned on by the clutch interlock switch on the clutch pedal.
You can check for power on either relay, the inhibitor relay or the clutch interlock relay, with a test light. Pull the relay out, have the key held in start position. Now both relays would have power showing on two terminals where the relay plugs in. If inhibitor relay only has power on one terminal or no power at all, either ignition switch is not sending power or fuse D is out or the inhibitor switch is not working right. If clutch interlock switch does not have power on two terminals either the ignition switch is not sending power or the clutch interlock switch is bad.
If the relay checks are good, check for power on the small wire to starter with key held in start. If you have power there, the starter solenoid must be out.

Yes there is a starter relay in the wiring diagram.
The yellow and lite blue wire goes from the starter solenoid to the relay. There is a yellow wire from the relay to a 30amp fuse. The key switch powers thru the neutral safety switch and then powers the relay on.

The dash lights coming on doesn't mean your battery is strong enough to turn the car over. That being said, I would lean towards the starter also. I would just have the battery chkd out before you go replacing other things just to be sure. Good luck to you.

Take a test light to the small wire on the starter and have someone turn it to the start position. If the light lights up and the engine doesn't turn over, I would suspect the starter itself. If the test light doesn't light, try it in neutral and repeat test. If it still doesn't light up, it may be the PRNDL switch or the ignition switch itself.

Just concentrate for the moment on the battery, starter, and ignition switch. Your 92 jeep is a complicated marvel, but all you need to do is get the starter turning the engine for all those automatic things to happen and turn it back into a mobile jeep.
As long as you have battery power at the switch, and even bigger battery power at the starter, and you connect the three of them together...instant bingo, the engine starts, or should if everything is working right. If I understand you right, you do have power to the switch, just nothing in start, or you 'd have noted that. Check engine light on, and warning lights and gauges work, yes? Then the switch is working that much, anyway. The switch start position should transmit power to the starter on the small wire that is on the starter. You can check for power there when the key is held in crank (start). Easy to check with a meter or a testlight. You stated battery and starter are good, so if the cable connecting them is good and tight., your problem may be in the start circuit from the switch to the starter (starter solenoid, to be exact). So it's ign. switch to starter relay to starter (solenoid).. Check for power on that path. Quick check at starter relay: pull the relay out (in the power distribution box) it connects to 4 terminals.. One terminal should be hot all the time. check with testlight. If nothing there, check the 40 amp fuse or battery connection or wiring. If one is hot, then have someone hold the key in crank. Now 2 terminals should have power. If not then ignition switch or the neutral safety switch or the wiring between them needs diagnosing. If two terminals do have power, try another relay. maybe yours went out. If a good relay doesn't help, then starter solenoid or starter or your wiring needs further troubleshooting. Hope you find it, over...

First thing to do is look at the dash lights when you put the switch in START position. If they blink or dim, you may have a faulty battery or poor connectivity.
Next try jumping the starter. You'll find a starter jumper at almost any auto parts. If you get no result then the problem is with the starter, If the starter turns then the problem is in the starting circuit from the switch to the starter relay.

Clean your battery terminals take them off and clean them. Turn on your headlights if you turn the key and nothing happens and the lights are bright hit the starter with a hammer try it a few times if it starts you have a bad armature in the starter. If nothing happens and the headlights go out or dim the battery has a bad connection or is bad. Check under the hood on the right above the battery the relays one is the starter relay switch it out with one of the extra ones in there. Put a volt meter on the battery just because you charged it doesn't mean it is 12.6 just sitting and not below 10 volts with key on

Definitely a starting system problem. That sound you hear in the back is the fuel pump.

Starting system circuit components (besides wiring harness)
1. Ignition Key Switch - closes the starter relay control circuit to energize the relay and close the contacts inside the relay (you can check this by locating the relay in the underhood fuse/relay box and have a friend turn the key while you listen/feel if the relay 'clicks'
2. Starter Relay or its Contacts
3. Starter Solenoid - pushes/pulls the starter gear into the engine's flywheel ring-gear and its contacts close to provide battery power to the starter itself.
4. Starter itself - poor brush contact with the armature's commutator will prevent the starter to spin/turn

If you are 'Electrically' talented, you can check to see if you're getting battery power to the starter solenoid by using a simple 12 volt test light/bulb. Connect one lead to the starter solenoid terminal and the other to chassis/engine ground and have a buddy turn the ignition key - if the light turns 'ON', then you know that particular circuit is OK but the starter solenoid is not energizing to push the starter gear into the flywheel ring-gear to turn the engine over. If it doesn't, the problem is elsewhere (wiring, starter relay, starter relay control circuit, etc).

You can have a qualified Hyundai Service Technician properly diagnose your vehicle. OR, you can Go to www.hmaservice.com and register your vehicle by VIN #.. you'll then have access to wiring diagrams, shop manuals, service bulletins, illustrations, etc.

The security system does not disable the starter, it only disables fuel to the engine. In the fuse block under the hood, locate the STARTER relay. Place your hand on this relay and see if it clicks while someone else turns the key to start...if the relay is ok then the starter itself is faulty. Good luck, i hope this helps.

I have a 93 pathfinder SE 4X4 automatic transmission that had full power, new starter and battery. Sometimes it will start and sometimes not. By passed starter relay by adding push button start and it started 6-8 times then nothing. I finally figured out that it was the neutral safety or inhibitor switch on the 4 wheel drive shifter. Just an idea..good luck