Splicing Different Gauge Wires?

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Splice away.

Wires don't know they aren't the same size.

12 is 12v. The question is going to be about amperage. How much of that 12v will the wire have to handle. Water is water but can a stream flow what the Mississippi River flows? 

Regardless of what I am connecting at the moment I know that a cig lighter can have other things easily attached. So I would build at least a 15 amp curcuit, it would begin with a 15 amp fuse and attach to a larger amp circuit. 

You need to study the circuit you wish to connect to. You need to know how much load it currently handles and consider the load of the new item. You need to know what drives it, today's cars even the interior lights often go thru a module and mistakes could be costly. 

Often adding such circuits is best done by adding a relay. The relay is a little guy telling a big guy to do the job. It takes very little from the lighting circuit as a command to turn the relay on. The relay attaches larger wires together to complete the new circuit. The relay can connect the new part straight to battery and not stress out any original wiring. This is the safest DIY approach. I'm sure there are many instructions to writeup a relay online. They are cheap too. 

The lights must not require much current if they came with 24 gauge wire. If 24 gauge is large enough for the lights, completing the circuit with 18 gauge is fine. 

You'll probably blow fuses or over heat the 24 awg wires. They are designed for low current circuits, and a lighter draws heavy current.

If you use the outlet for operating a radar detector or other low current accessory, it would probably be OK, but never use it for a cigarette lighter.

Look for a different powered circuit containing 16 or 18 awg wires if using it for lighters.

NO! Different gauge wires are designed to carry different loads of electrical current. A 24 gauge wire MAY overheat and cause a fire if connected to a high amperage circuit like a cigarette lighter.

You're good to go.

The lighting won't draw anywhere near the current limit for 18 gauge wire.

Yes it should be fine