Hello my name is Frank. I bought a new starter and installed it. I turned the key and I got the same one click noise as before. I had 2 options...

The first thing I would check is the earth from the engine to the body/battery as the power is obviously getting to the starter on cranking. The burning wires would be the result of the power trying to find an alternative way back to the battery making the circuit. Listen carefully to the starter when turning the key, make sure it does not run on when you release the key from the start position. 

Your problem is with your battery or the battery cables. It takes very low amperage to opreate the ignition switch however it take very high amperage to crank the starter motor. You will get most all electrical components to work with a week battery or bad cable connections but the starter will not.

Starter issues are simple to diagnose. Attach test light or volt meter to the small single wire terminal on the starter. Simulate starting the vehicle with key. If test lights or get battery voltage at meter and doesn't start than starter is defective. 

Seems to me that you grounded out the ignition wire somewhere. When you cranked it the ignition wire shorted to ground thus melting the wire.

You can simply start over. Or replace the melted wire.

Just get a basic relay. Bosch. Hook it up yourself. Make sure the starter and engine block are clean. Ensures a good ground.

I could be possible that the new starter was defective.  Simply check for continuity from ignition terminal to the starter case. If you have continuity that would be the cause of melted wires.

Hi Frank, I'm going to assume you have a 2.3L engine.. at the Starter solenoid is a Red/Light Blue wire that's trigger from the ignition switch to energize the starter motor, which is powered up only when the  vehicle is in park and if it's a manual transmission while the clutch is pushed in.. Using a test light unplug the Red/Light Blue wire and turn the key as if you are cranking the engine the test light should light.. if this tests passed... Try jumping the bigger terminals together with a jumper wire or screw driver make sure the vehicle is definitely in park and the parking brake applied the start motor should energize if so your problem is the starter solenoid.. if that doesn't remedy the issue make sure that the negative battery cable is securely attached to the engine corrosion on either end of the negative will cause higher than normal resistance and cause stating system faults.. hope this helps

replace that starter relay, let me know soon

Hello and welcome to ExpertHelp.com! My name is Relly De Jesus and I'm going to do everything in my power to answer your question to your full satisfaction!

Before we get started, I want to remind you ExpertHelp.com is an independent professional support company. We do not have any affiliation with any brand nor are we paid to provide support on their behalf. However, we have helped hundreds of customers with similar questions and believe we can help you too. OK, let’s get started! :)

I’m available to chat now. Please let me know that you are ready by posting a response. I’ll leave my chat session open for the next 15 minutes or so and wait for your reply. If I don’t catch you this time, please respond with a few times (including your timezone) that work best for you and we can connect then.

Hi Frank

How are you doing? I am suspecting that the relay is loosen which may have cause the issue. If seated tight I believe you'll be good. Please let me know what happens tomorrow when you start it. I'll be standing by and good luck!

Thank you

Relly

Hello Frank

How are you doing? Hope the truck is doing great as well. Please let me know of the result.

Thank you

Relly

Hi Frank

How are you doing? Any update on the truck? I'll be standing by for your response.

Thank you

Relly

Hi Frank

How are you doing? Any update on the truck? I'll be standing by for your response.

Thank you

Relly

Hello Frank

How's it going?

Relly

could you leave me your text an ill get back to u. this is truly new to me answering guestion

I believe this Q has already been expired Randy