First off, congrats on your first purchase...which may lead to many more.

Second, consult this - http://www.pinrepair.com/em/index.htm

This is also a decent video on learning EM repair. Never mind the bullshit spewing.

Third, at least it's a single player. Only one set of score reels to deal with.

Wait for some other more experienced guys to chime in but here's my thoughts. So what you're saying is the game will begin its reset sequence when the start button is pressed but the score reels won't reset unless you hold down the score reset relay in the back box. Or, are you talking about the game reset relay on the mechanism board in the bottom of the cabinet?

Until you know more keep your metal tools away from the game with the power on. I don't worry about shock with my fingers when I'm just around the relays, coils and lamps. Just stay away from the line voltage and the transformer especially.

Do you have a little metal file? You'll need one. You can get cheap one's at Harbor Freight but I like this Nicholson one. Hold's up well. Although, remember, you're just cleaning the contacts. Not filing them down. So take it easy with the filing.

http://www.amazon.com/Nicholson-02375N-4-Inch-Tungsten-Point/dp/B007C6OKB0/ref=sr_1_1

PB Resource is also a great place for EM repair parts. Here's a link to the tools page. You'll want a "L" Contact Adjuster as well.
http://www.pbresource.com/tools.html

Triple Action looks like a cool game. Never played one. I like orbit shots so I'd probably rubber band the upper left gate so a good right orbit shot will loop around back to the flipper. Does it have DC powered pop bumpers and slings? The coils for them would have orange wrappers as opposed to blue ones for the AC powered coils. Williams DC powered coils from the 70's kick pretty good. Just not sure how early they started using them.

Do the score reels spin at all when you press start? If not then the score reel reset relay in the back box may not be firing. Could be a couple reasons. Look for broken wires between that relay and the score motor (big thing with nylon cogs in the cabinet) which pulses the relay to fire when applicable. Might need to clean the contacts and check to see if they are making proper contact when the motor runs. With the power off you can turn the score motor cogs and watch them lift and lower the switch stacks. Now unless someone really screwed with the game you should be adjusting only a very small amount of switches. Nonetheless, all of them probably need cleaned.

FirePower - (awesome game I own one too) is right. You should inspect and clean every contact in an EM. Then you know the game inside and out. My EM's have all been barn dwellers so this was a necessity. You are lucky to have a mostly working game out of the box but if you're gonna get into pinball learn the game by touching, cleaning and examining every part.

The score reset relay will fire and hold until the score reels find zero. Maybe you just got a score reel that needs a switch adjusted. Pull one of them out. Little metal prong at the bottom of each reel assembly. Manually activate the score reel with your finger. Scores 1 through 8 should have the switches in one spot. Moving from 8 to 9 will be your carry over and 9 to 0 will find its reset. Make sure all of the reels show this. Watch how the switches move as you manually activate the score reel.

Ok. I just watched your videos. The score motor starts running as soon as you turn the game on. Looks like the game over relay (in the cabinet) is not properly activating. Find the game over relay in the bottom of the cabinet and manually activate it as the score motor is running. If it stops then you probably have an issue where the score motor switch is not pulsing the game over relay to fire. As always look for broken wires first.

Going by the video something is not resetting as soon as you turn the game on and before you even press start. Does the game have steppers in the bottom of the cabinet? Manually reset them all before turning the game on. Small coil is for resetting. Large coil is for advancing. Keep manually firing the small coil till the stepper won't go any farther. Maybe the reset arm on the one of them is binding or sticking. This is a common problem with Williams EM's. Would just need to be removed and cleaned.

At this point I would trace the coil wiring from the score reset relay to the score motor on the bottom of the cabinet. One of those many switches stacked on there pulses that relay during the start up sequence. Either it's not making contact due to being slightly out of adjustment or the contacts are dirty and need to be cleaned. You can use a q-tip (I buy boxes of the cheap ones at the dollar store) and some 90+ percent alcohol and wipe the contacts with the game off till you get a file. It might be tough to get the tip in there so take it easy.