@fishnjeeps - Welcome to the forum.
You asked:
So, what spare parts do you guys suggest I carry for long distance trips? or how does one handle more serious breakdowns?
You ride a nice bike. The 92' Kawasaki Vulcan 88 is a pretty reliable, shaft drive, water cooled, carb'd bike with cast wheels/tubeless tires in the cruiser style, (for those unfamiliar with them). For general purposes, the items others have recommended will do well in taking care of minor issues. A flat tire shouldn't be a major issue with tubeless tires. From your forum pic it looks like you added a windscreen, which helps reduce fatigue during longer rides. No doubt helps you commuting too, with the weather you see in IL!
Most riders will not carry major spare parts. But, knowing your bike and what things tend to go wrong goes a long way to understanding what parts to bring and what you can, and are willing to do, by the side of the road. I used to carry extra head gaskets for my '40 HD Knucklehead, for example. But while that took 30-40 minutes to do by the side of the road, it was a simple task that could be done with basic wrenches. And something I had to do now and then on that bike with it's paper head gaskets. Technology marches on.
So, considering your previous failure on the road. What tools and items would have allowed you to do a field repair and continue riding? Would some thick wire and pliers allowed you to wire the exhaust back in place enough to continue? Or would it have gotten you to an auto parts store where you could have gotten some generic materials like muffler wrap that would have allowed you to do a field repair sufficient to continue your ride?
Thinking outside the box and being willing to do a repair that will get you going, with out necessarily being the permanent fix is part attitude, part mechanical ability and part simply a matter of what you have on hand or can find in the next town/gas station, etc.
A rider during the IBR once took his clutch apart by the side of the road and re-assembled it with a piece of metal from a coffee can that he found lying around in order to get his burned up clutch just a bit more friction and allow him to continue. And yes, it worked and he did finish. That rider that carried a spare BMW final drive didn't need it, (actually there have been at least two that did that during IBRs), but had practiced doing it and could do it in about 20 minutes by the side of the road. That was a known failure issue. One IBR, several riders DNF'd for failed final drives, so it was not a choice made lightly.
I crashed once about 800 miles from home, (Ice). I lost both mirrors and my wind screen and bent the controls out of whack. I was able to pull my tools out and adjust the controls back into rideable condition. I should have then ridden to the nearest auto parts store or dollar store and bought some kind of mirror, even a cheesy makeup mirror, and taped it to what was left of one of my mirror stalks. I should have found some cardboard or a piece of plexiglass/lexan at a hardware store and taped that to the front of the bike so I had some form of windscreen. Instead, I rode the 800 miles home that day/night w/o mirrors or a windscreen and I'm sure you can appreciate how cold it got in Jan crossing the mountains. Riding in traffic w/o mirrors made for a lot of head checks!
I carry the tools I need to access general areas of the bike, remove the wheels, (because it's a LOT faster to get a tire replaced at a shop when you carry in a loose wheel, as well as cheaper, and sometimes tire damage can't be safely repaired), a roll of high quality HVAC grade duct tape, (the cheap stuff peels off at highway speeds), a roll of electrical tape, (used as a strip across the top of my helmet visor when riding into low angle sun, as well as repairs when needed), a roll of yellow gas seal tape, (like teflon tape, except gasoline proof and thicker, available at Home Depot and similar stores in the plumbing section), a couple pairs of nitrile gloves, a spare headlight bulb, turn signal bulb, (my brake light is LED or I would carry one of those too), and know how to access the bulbs all over the bike. (yours are mostly just screwdriver tasks) Some cops will make you park the bike and camp by the side of the road if you don't have a headlight or brake/tail light at night or even dusk. A spare key is a good idea too. Sometimes they break, or get lost in unforeseen ways. If you don't have room for an entire roll of duct tape, take a pencil and start wrapping tape around it. 20' or so doesn't take up as much space when there isn't a big center tube. I have helped tape together several sport bikes with damaged plastic using my roll of duct tape, but I have a good place to carry it that doesn't take up extra space.
You can usually find odd things to help with repairs where ever you are. A rock to prop the bike up when you pull the wheel off, for example. The Vulcan doesn't have a center stand, which adds to some of the challenge. This is just a way of saying you don't have to carry every possible thing on the bike. You don't have the room. That said, there are some pretty cool tools for propping up a bike with no center stand and they don't take up much room in the bottom of the saddle bag.
Doing preventive maintenance helps avoid issues. Most LD riders running shaft drive bikes will change their final drive oil a lot more often than recommended, for example. This helps longevity, but also means you catch problems before failures when/if you see lots of metal in the drained oil. It's cheap and easy to do as well. I do it about every other engine oil/filter change. Some do it every oil change. If your wheel bearings are over 100k old, you should consider replacing them before they fail. Japanese bikes tend to go a long time before bearings fail, but it's not a repair that you can easily do by the side of the road, even if you have the bearings. Inspect your brake pads before you go on a long ride. Know how to drain your gas tank in case you get bad fuel. or in a fatigued state, accidentally fill up with diesel, (it happens, even to experienced riders). I'm not of the mindset to replace batteries every two years. I typically get 7-10 years out of a well cared for AGM bike battery. I do ride often and use a battery maintenance charger. Most of the time you get some warning that your battery is getting tired, but more often they die early because they were mistreated, like full discharge.
Read the Archives of Wisdom on the IBA web site. On LD rides, carry water where you can drink while riding. Not as critical in Winter, but very important the rest of the year. Dehydration slows your thinking process down and you make more mistakes. I know the Vulcan is on the minimalist side and you can't carry as much. Not sure if you have the ~3 gal or 4.2 gal tank, but knowing your range and how far you can get on reserve is important and makes it less stressful when you have to turn that petcock over to the R.
This stuff is supposed to be fun. If you're not having fun, there is always another day. Good luck with your riding and Saddle Sore/certificate rides.