I have an electric soldering kit. I've soldered a few times so I'm up for trying it.

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So basically I'm just running wires the length of the distance for where I don't want the strip illuminating.

It there a particular gauge of wire I should be using?

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What about using solderless connectors like this?
https://www.amazon.com/Strip-Connector-Solderless-Accessories-Provides/dp/B07RG6T1FV/ref=sr\_1\_7?crid=3A8YFAKRUJ4S&keywords=led+terminal+extenders&qid=1679522659&s=hi&sprefix=led+terminal+extenders%2Ctools%2C85&sr=1-7

I used this kit to make angles https://www.amazon.com/gp/product/B07RG6T1FV

You can sodder or you can buy crimps (https://www.amazon.com/dp/B07RG6T1FV/ref=cm_sw_r_cp_apa_fabc_rS7aGb8KTSBGZ?_encoding=UTF8&psc=1). But for under $30, you can get a controller that comes with LEDs. I think it's more cost effective the second way.