There are several factors that affect combat: Your weapon (surprise!), boots, rings, healing food, buffs, maximum health, and experience.
Weapon
The better your weapon is, the faster the enemy(s) is killed, the less chance you have of getting hurt. The others have covered it pretty well, all I would say is that while the next best sword is definitely nice to have, by no means is it required. The Lava Katana is already a formidable weapon.
Boots
Boots provide two benefits: Defense and Immunity. Defense reduces the amount of damage you take, and Immunity reduces the chance of being affected by a "debuff" (such as being slowed down by a Slime). Different boots have different values of each; the Space Boots are pretty solid with 4 for each. The only way to get boots better than this is in the Volcano Dungeon itself. Make sure to open the chests and you'll eventually get some snazzier sneakers.
Rings
Rings provide unique bonus effects that assist in not just combat, but other tasks too (such as a Glow Ring, which spreads light around the player 24/7). You have two ring spaces. My suggestion would be that you craft a Glowstone Ring (5 Iron Bars, 5 Solar Essence, recipe from Mining Level 4), which provides both the glowing effect AND the magnetism one (collect items from further away). This is incredibly useful; once you have one you can't go back!
For the other space, why, the answer is simple: another Glowstone Ring! The effects stack, and having two makes it that many more times convenient. There are other, more unique rings that you can get later in the game, but for now I wouldn't bother with them. You'll see why soon! ;)
Healing Food
The other three have both recommended Cheese (a great but rather expensive option), so I'll provide some alternatives in case you don't already have easy access to it.
Salads, which can be bought from the Saloon for 220G each, also heal a significant amount of health. The nicest thing about them is that they simply require money, which you can get any number of ways.
Sashimi doesn't heal quite the same amount as Cheese or Salads do, however it is significantly cheaper. I know that you have the kitchen (it's required for marriage), so all you'll need is 3
s with Linus for the recipe, and then you can turn any fish into Sashimi at a 1:1 ratio. I recommend using Mussels, Cockles, or Oysters, either from Crab Pots, or the Mussels Nodes that spawn along the south shore of Ginger Island West (
the farm), as they are very cheap.
Buffs
Most cooked dishes provide you with one or more bonus effect(s), called "buffs"; these can be discovered in the tooltip of the dish. The most useful ones for combat are Speed and Defense. As
@Ereo said, Speed is useful for outrunning enemies, but also for making faster progress through floors, and Defense has been covered already. Three useful dishes for mining/combat are
Spicy Eel (+1 Speed, +1 Luck),
Crab Cakes (+1 Speed, +1 Defense), and
Pumpkin Soup (+2 Defense, +2 Luck). Make sure to stack some Coffee on top (drink-buffs do NOT overwrite active dish-buffs, unlike eating another dish) for an extra +1 Speed, preferably Triple Shot Espresso, as it lasts longer.
Maximum Health
The capacity of your health bar determines how much "wiggle room" you have before you must top up on health with healing food. This is determined by your Combat Level; you can view this in your Skills tab. The higher your Combat Level, the larger your health bar. To increase it even further, choose the Fighter profession at Level 5, and then Defender once you've reached Level 10. However I would recommend not taking this second step, as the increased attack damage of Brute is preferable IMO.
Experience
Obviously, with practice comes more experience. And with more experience, you'll be able to better predict which way monsters will move, time your swings with more precision, and know what a dangerous situation looks like before you're in the thick of it. I recommend spending as many free afternoons as you can in Skull Cavern, purposefully interacting with enemies instead of concentrating on gathering ores. You'll end up with a bunch of handy loot this way too!
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I also have some useful tips specific to Skull Cavern & the Volcano Dungeon.
Skull Cavern
Skull Cavern is the best place to find the valuable Iridium Ore. It spawn in nodes like other ores, but can also be dropped by the Purple Slimes and some other monsters that start to spawn as you go deeper. It is needed for the final tier of tools; you'll have to bring 5 bars and 25,000G to the Blacksmith in order to upgrade. It's incredibly useful on the Pickaxe, as it lets you one-shot the rocks in SC, and brings the rocks in the Volcano Dungeon down to 2 hits; make sure to use your first few bars on this. It is also needed for other lategame recipes, but I won't get into those here.
The deeper that you get, the more Iridium that spawns. However, unlike the normal Mines, there is no Elevator. This is why uncovering ladders is a priority in the early stages of a SC "run", and also why it's best to dedicate full days to your runs if possible. Make sure to only go on positive luck days, or else the lack of ladders can quickly become frustrating. Watch out for shafts, these will let you skip several floors in one go.
Once you're more comfortable with SC, you might want to consider checking out the
Skull Cavern guide that I made! It covers every aspect of Skull Cavern (preparation, strategy, and mechanics) in depth. But be warned, there are major spoilers in it, and it's also a
little bit long. Y'know, only 11k words or so. ;D
Volcano Dungeon
Like Lew said, just run run run! The Volcano only has ten floors, and there's a fun surprise at the top. Once you've made the trip once, you won't need to do it again, but there is still valuable loot to find within it. Press every button that you see, ignore the Magma Slimes, and you'll do fine. Just make sure to pick up all of the forageables, as they're really useful later! You'll also probably need to dedicate a full day to it the first time. Experience makes a massive difference, so don't worry if you're not making much progress at the start.
But DO worry about making it back home in time, unless you're happy to lose 2.5k G every time! Because there's no ladders, you'll either have to run all the way back to the start, or use the purchasable escape chute at floor 5.
Mining is really fun once you get the hang of it! Sorry if this a little overwhelming, but I hope my 1200 words help make a difference!
Have fun! :)