What settings do I need for night photos with my Nikon D3100?

Last night I tried some night time photos and wanted to really capture some nostalgic scenes. I chose a spot diagonally across the street from a classic looking restaurant with neon signs so it would look like it was from the 50s and waited until cars passed by to capture their tail lights going by. I slowed the shutter speed and had my camera on a tripod. While the car lights looked great and the street lighting did too, the restaurant's lights were still too bright no matter the ISO. I tried a shutter speed of 13 and went down to 3 and still ended up with the same results except I had great looking smearing of taillights to show cars whizzing by.

In order to capture neon signs at night and be able to read them clearly in a photo, what settings should I use?

You need to adjust the lens aperture to assure that the lights in the restaurant are not overexposed.

Once you have the correct exposure for the restaurant lights, use a reciprocity chart to figure out what combination shutter speed, ISO and lens aperture you need to get what you want.

There are NOT magical "settings" when using a fully adjustable camera

What do you mean by "a shutter speed of 13"? Did you mean 1/13 of a second or did you mean an APERTURE of f/13. You need to get a handle on your terms or it makes it very difficult for us to answer. If you meant "aperture", then going down to f/3 would make the lights brighter. Similarly, if you meant shutter speed (although I've never seen a camera with a shutter speed of 13 sec or even 1/13 sec), then you have to clarify whether you went from 13 seconds to 3 seconds or 1/13 sec to 1/3 sec.

Basically, try this.
Set your ISO at 100. Pick a shutter speed, let's say 5 seconds. Set your aperture to f/11. Shoot the image. If it's too bright, you'll have to do one of two things. You'll either have to go up to f/16 or f/22 or higher on your aperture or you'll have to decrease your shutter speed to 3 seconds, or 2 seconds, etc. The faster your shutter speed, the shorter the lengths of the car taillights because your shutter's not going to be open long enough to create a long trail. There's always a trade off.

The fact that you're using a D3100 has no bearing whatsoever. The principles are the same regardless of the camera. There's one exception. Check your menus. Some Nikons have an automatic override. This means that no matter what you set the ISO on, the camera will pick the ISO it thinks is best anyway and it's usually wrong.

Finally, you need to get a good handle on the relationship between ISO, shutter speed, and aperture.

If you take an image and want to take it with different settings but you want the image to be exposed exactly the same, you have to know this. Here's an example:

ISO = 100 Shutter speed = 1/125 Aperture = f/8
ISO = 200 Shutter speed = 1/250 Aperture = f/8
ISO = 200 Shutter speed = 1/125 Aperture = f/11
ISO = 400 Shutter speed = 1/500 Aperture = f/8
ISO = 400 Shutter speed = 1/125 Aperture = f/16
ISO = 400 Shutter speed = 1/250 Aperture = f/11

All of these will create exactly the same exposure. If you go up one stop on your ISO, you have to go down one stop on either shutter speed or aperture. If you go up one stop on your aperture, you have to go down one stop on either ISO or shutter speed, etc.

First, "Froll" apparently has never heard of the "Bulb" setting that allows you to keep the shutter open as long as you want - even 13 seconds.

Second, I agree with him that you need a better grasp of how to express things such as shutter speed. 1/200 sec., 1/2 sec., 10 seconds, etc.

An old "Day & Night Exposure Guide" I've often used in the past recommends this setting for neon lights, using ISO 100.

f4.0 @ 1/30 sec.
f5.6 @ 1/15 sec.
f8.0 @ 1/8 sec.
f11 @ 1/4 sec.
f16 @ 1/8 sec.

How accurate is the Guide? Well I used ISO 200 and f11 with an exposure of 30 seconds for these. You be the judge.