I'm not new to resource management games, but I do have a hard time learning the best way to do something. I've tried a couple of different methods of getting cash but now I'm at the 10,000 mark and I am not having a good time trying to figure out how to survive it. Any tips or tricks I should try? Or shall I abandon all hope?

Get 500 reputation point in the 3 major guild (Fighters, Workers, Mages). Do their quest for the Mayor election. And after the election event the debt will be gone.

There are three ways to beat the debt I know about, I haven't played the newest version yet though I have wanted to just have had other things to do.

The method I've used before is to focus entirely on resource collection starting with fishing, then unlocking woodcutting, stone, ore, and smithing in that order and only acquire new slaves when you have something for them to do trying to get ones that like fish. Sell all but the food you need to keep your household fed and all resources you gather that aren't being put towards upgrades, doing this I was able to finish the entire 100k debt around day 60 of 100, a few of the early weeks might be tricky but should be easy enough once you get past them. While doing this I also ignored 90% of the guild quests and only started combat excursions once it was impossible for me not to make the cash goals.

The second method would be as vadi says, rush reputation as fast as possible to advance the main story and get the debt "frozen".

The third method I've heard people try but haven't attempted myself is to focus on combat, not sure the totality of that strategy.

I've succeeded with the combat-focused strategy and paid off the entire 100k with weeks to spare.  It is harder than it used to be due to the changes in the loyalty and dating systems, but still possible.  

General tips for this strategy:  

• I basically have two types of combat characters--fighters (high-Physics) and mages (high-Wits).  Typically I start with my MC as a fighter and my starting slave as a mage.  Optimize your starting factors accordingly, but don't completely neglect the MC's Charm and Sex factors.  
• Growth factor is very important for combat characters.  
• I typically give fighters the Archer class first--it gives you Wind Attack, which hits twice.  Follow that up with Worker (+Physics and HP, and significantly lowers the amount of time required to gather resources after finishing a dungeon).  After that, various combat classes, optimizing for Physics for high damage output.  
• For mages I generally start with Apprentice, which gives +10 Wits and a basic magic attack.  Next comes Thief (gotta have someone to pick locks/deal with traps, and Wits and Evasion are nice).  After that, generally focus on classes that boost Wits.  Take Acolyte reasonably early for healing.  
• Start on dungeons early (there's a starting set of them with locations already known).  At the very beginning your characters will be quite weak.  Look for any "Trouble Solving" guild quests to start with.  After those (or if there aren't any), go take on a Goblin Cave.  Buy out the Healing Shards from the shop until you have other ways of healing.  Note that you can pause to buy things from the shop in between dungeon encounters--or, if need be, you can just rest/pass time at the dungeon (which allows you to heal slightly, and wait for the shop to restock).
• Groves are the next-easiest dungeon type overall, except the end boss is a Troll that hits *hard*.  Teach a mage the Lightning ability before taking one on, and make sure your front-row fighters can take at least one solid hit--but the Troll is very susceptible to the stun effect of Lightning, so stun-locking is pretty effective.  
• Once you've leveled up a bit, you can start on Bandit Dens, which will be your go-to dungeon type for a while.  They give pretty good EXP for leveling up characters, and can get you new characters--I decide to keep or sell each bandit leader based on their stats (*especially* Growth Factor), as well as how much room I currently have in the Mansion and how much I need a new worker (I'm much more likely to take in a mediocre bandit leader in the early game).  
• Do dungeons.  Constantly.  The gold alone from beating a dungeon is usually more than it costs to buy the dungeon.  
• For equipment, focus primarily on offense (ATK for fighters, MATK for mages).  Unless it's *right* before a payment is due, don't be afraid to invest money in good equipment if it happens to come up in the shop.  
• Be aware of elemental weaknesses/resistances.  (Right-click on the enemy portrait.)
• Especially for the first couple of payments, it'll probably be rough.  Know when the payment is due, and you may have to sell a bunch of stuff at the shop the day before.  That's ok.  Stuff is replaceable.  (Also, raw materials sell for a lot.)
• Mid-game, Bandit Forts are good for EXP and loot, and also let you capture the boss at the end.  But make sure you have a plan for dealing with the Bandit Assassins, since they can frequently one-shot your mages.  (Generally, I wait until my fighters are leveled-up enough to have higher Speed than the assassins, and are armed with bows to be able to hit the back row.)
• Gradually upgrade your Mansion.  Late-game, when you have level-3 Forge/Tailor/Alchemy, invest in making top-tier Weapons/Armor.  The different types of weapons and materials have different tradeoffs, so there isn't a single optimal weapon type.  But generally make the "advanced" armor and weapon types from the highest-grade materials (e.g. adamantium longbow, dragonbone master staff, etc.).  I often go for Advanced Medium armor on my fighters (instead of heavy) because the Evasion boost is very good (especially if you add in Air Crystals...).  Obviously, you'll probably have gone through all the dungeon types a few times in order to get this far.  But once you're equipped with these on a full party, you can repeatedly chew through Fire Depths pretty rapidly, and you'll be able to accumulate 100k worth of mithril alone.  :)