wiring in a relay is not very hard. sometimes the hardest part is taking apart the trim to get to the wiring! basically what you'll be doing is cutting the + lead to your lighter a few inches away from the socket, adding in a relay (which works just like a switch but rather than a human toggling the switch, you get electricity to do the toggling for you via a tiny electromagnet), and then hooking up that relay to a key-switched circuit nearby to act as the trigger. assuming that pic you posted is of your car, then the wiring for your heated seats would be the perfect trigger circuit to use, since it's already right there, it's most likely key-switched, and thus you wouldn't have to tear apart the entire dash to find a key-switched wire to tap into. even if your car doesn't have heated seats, the wiring might already be there to tap into - just need to test using a voltmeter or test light.
you might want to consider going to a shop that does stereo or alarm system installs, and ask them to add a relay so that your lighter is only live when the key is on. should be relatively easy for a shop to do, and hopefully they won't charge you more than about $50 to do it, parts included. i could do it in about 10 minutes once i figured out how to take apart the trim pieces to get where i needed. unfortunately you're a bit far from houston, tx, or i'd offer to do it for you.
if you've never done electrical work before, and thus don't have any tools or parts, you might end up spending almost $50 buying everything you need to do it right, and then you'll have extra wires and connectors left over, taking up space. but then you'd also learn a valuable lesson in basic electronics, too, so that might be worth it for you. i started out with simple things like this, too - wiring up a power circuit for a PC i was building into my car to have MP3s through the stereo. that was back when the biggest portable mp3 player you could buy had 128 MEGS (not gigs!) of storage, and a cd deck that could read MP3 CDs was yet to be invented, or was still super expensive. meanwhile i was installing a retired K6-200mhz with 16mb ram and win95 on it, along with a 17gb drive full of music.
that was almost 15 years ago. now i just use my android phone, but it was a neat project and i learned a lot from it.
if you do decide to do the work yourself, STAY AWAY from any wiring wrapped in bright colors such as yellow, orange or even blue. these are often for critical systems like airbags or if it's a hybrid/EV, they might be high current battery lines that will kill you if you handle them wrong. don't let that scare you though - just don't cut or tap into them and you'll be fine.
