Reverse light, headlight, running light, and turn signal clusterf

Alright, so my latest problem with my car is clearly something with the electronics, although I'm not sure if this should still belong in the "Newbie" section. Hopefully this is the right place.

Anyways, here's the problem. My turn signals, running lights, reverse light, and front right headlight are not working. I know that all of these things worked before I swapped tails, which caused some initial problems. However, I've now installed totally new bulbs, the harness is good, and I have a new tail light sensor. Both brake light bulbs work when the brakes are applied. The reverse lights do not work, but interestingly when I put the car in reverse, the dash lights along the bottom come on. Not all of them, but most which doesn't make any sense. In addition, the turn signals don't work at all, and trying to use them results in a very fast clicking, as if there is a bulb out somewhere but even faster. The hazard lights, though, cause only the right turn signal to blink at the correct speed, which is odd because the front right headlight is out. Which brings me to the front right headlight, which I replaced with a new one but still won't work, and once again the wiring looks good.

So, my plan is to buy a multimeter and some fuses, and try to find out if I have a blown fuse or bad wiring somewhere, though I'm not sure exactly where to start. I'm just posting here to see if anyone has any idea as to whether or not all these problems are connected, or if there is an easy way to test some of this.